ak a (fF | =(). Now, as the medium under consideration (a mixture of gases, suppose, or a solution of salts) is composed quite arbitrarily of several simple constituents, each of which is characterized by e, €', x, and a, and we do no assume that these constituents 2 ally exert reciprocal action upon one another, the fore- 342 Prof. EH. Ketteler on the Dispersion of Light going total condition resolves itself into particular condi- tions. Hence we shall have Qe ~ € -- (ey) — fone) oe = 0, /2 et @a—HP)s=0, wifeveinis Bearing this in mind, then the total result of the investigation hitherto is comprehended in the following most abbreviated form :— /2 seu ID peo, vale p Kl—e mp : 6. On account of the nature of the last two equations, it is of course impossible to bring them into the form n=/(A) and | therefrom deduce rigorously the properties of the true disper- sion-curve. Indeed, for a general view, what was said above respecting the approximative formula (9) is sufficient. It is thence obvious that, if we go through the middle point of a particular curve n of equation (11), we at the same time pass through an apparent inconstancy in the true dispersion-curve (an absorption-streak). If on the two sides of this middle line the p/;, of equation (11) has opposite signs, so that the particular vis viva m' yp’, on the right is added to the other terms, but is subtracted from them on the left, within the limits of the ab- sorption-streak the wave-length / and index of refraction n — become complex. Now I have shown, in my memoir cited above™, that in this case the reflected and refracted light is very elliptically polarized, and that in the transmitted wave, not merely do the ether particles suffer a sudden change of phase y, but likewise the corporeal particles undergo a different alteration of phase y’. If, now, the actual excursion of the particles of both kinds, within the limits of the complex zone, are denoted by p’,, po, it follows, from a generalization of the point of view there developed, that the existing equation pe 4 m (ry ty/—1y (a+b Eels mt +to/ —1)? divides into the two following — mp’ 2 2 yas 0 ; ; a?—l?’—-1=> shop? cos 2(y/—x); 0 (12) /2 / Yab=z2; a sin 2(y’—¥). 0 « * Verhandl, [4) vol. ii. p. 93, in Singly and Doubly Refracting Media. 343 If, then, for the nth zone, and for the limit-point G’ to the right of it, we have y’—y=0, the whole of the particular ex- cursions will, with the entry into the absorption-streak, simul- taneously undergo such differences of phase ; but these reach at the middle line only an insignificant maximum, and on the other side of it sink again to zero. Only for (p%, py), does 2(y’—x) rise considerably ; it reaches, at least with very feeble dispersion, at the middle line the value +90° (corresponding to +7/), and for the limit-point G” on the left the value +180° (or +3/); so that here, indeed, againb=0, and therewith the curve becomes real, but the sign of p’; in equation (11), in accordance with the hypothesis, changes into its opposite. Forming, lastly, the resulting ratio of the vires vive for the interior of the complex zone, we can put R’ 2m’ | Rem All the ealeulations can now be carried out for a substance with only one inconstancy in the spectrum (n,,=1), conse- quently with omission of the symbol of summation. We find, in the first place, for very small dispersive forces, a Pn = x 2 SOEs. taal ee =vD=4/ (1-2) =V/(1—ni)?, | : NA sin 2’ —x) = 4/ =, c08 yx) = 4 / m 0 ; Ps m0) =/(@—P— 1p +408. We then obtain, pursuant to the above convention :— Left of the middle. Right of the middle. Nin — 4/ Am— N—r of A—Ar aa on fe el SES (gps a’ —b ae Vee i? ale ,, Naa baba My [PN Noma, [WER Ls x Xm— rv", Wes NO m Designating now the expression a?—b’—1=N?’—1 as the merely refractive, 2ab as the at the same time absorptive part of the index of refraction n, the former diminishes as we pro- ceed from the right-hand limit G’ of the absorption-streak (at which it attains the value +4/D=n’/j—1) to the middle line. At this, a? —b’=N’=1, and sinks for the limit-point G” on the left still more, to 1—./D=n’%. If, therefore, we imagine the limit-points G’, G’’ connected with each other by a curve 344 On the Dispersion of Light in Refracting Media. constructed according to the foregoing law N = F'(\)*, this curve forms in a certain relation the continuation and complement of the till now isolated branches of the real dispersion-curve ; its ordinates increase simultaneously with the wave-length. The absorptive part, on the contrary, increases from the two limits to the middle ; hence everywhere the upper or the lower sign only is to be used. Passing, finally, from quite feeble to stronger dispersive forces, the consideration of equation (4) gives, on substituting . e xr . e . e in it = for 7, and solving it according to n?—1 instead of n, immediately : 2 of, ne + ea he =,(UesD= ate VAG +D-<) Stal) It divides for the complex zone into the two following, 2 @—P—1=—}(14D- x) RPM Ok AUN — cae A] Daa =) : YAY . ° m Pp 0 TT) ° : and gives, besides wp =%¥TD, more complicated values for 2(y/—x). For the middle line particularly this angle always becomes less than 90°. That. the processes here indicated never take place without absorption, is scarcely surprising. We have, in the occurrence of the difference of phase between the ethereal and corporeal particles, an implicitly communicated coefficient of friction; and the intensity of the absorption} will be exactly proportional to * Of course N must be distinguished from the actual velocity- or sine- : » sine : week . ratio, y= — = —— (for perpendicular incidence = a), whose difference o sir ! y—l=™ Q2, becoming variable, must be regarded as the actual refracting i) force. Cf. l. c. pp. 70, 85; et mfra sub § 16. + If, after Cauchy, we put the absorption-factor of the amplitude aw ab equal to e*, @= ren, for D small, , the abscissa z being taken vertically, we have, ps EO) mp—m'p'> ” where d denotes the thickness passed through, and 4 is only negative. I accept this fully, but conditionally—that is, rigorously retaining the receding signification, and without any connexion with the so-called ‘Ean eee Conf. J. c. p. 69. Mr. W. H.Walenn on Division-Remainders in Arithmetic. 345 sin (y’—vy), indifferently whether its sign (and therewith the . resulting elliptic polarization) is positive or negative. Under this assumption, the difference of brightness between the edge and middle of an absorption-band becomes greater in propor- tion as D is less and consequently the band is narrower. Moreover the preceding discussions have cleared up many formerly obscure points ; andif the approximative formula (9), with corresponding treatment, may be applied to the absorp- tion as well as to the refraction of composite media, on the other hand it can scarcely be any longer doubtful that the phase-difference (y’—) does not, like the “ coefficient of ellipticity,” diminish towards the interior of the medium, but much rather remains constant, so that in each successive stratum an equal quantity of regular is converted into irregular oscillatory motion. [To be continued. | XLUI. On Division-Remainders in Arithmetic. By W.H. Watenn, Mem. Phys. Soc.* RE GARDING the dividend, divisor, quotient, and remain- der in an ordinary numerical division as algebraical quantities, the operation may be put in the form of the equa- tion a ia o13 G b cand b, and » cannot be obtained from it without knowing c The expression c+ ; is a function of 7; it is also a function of as wellas b. Ifthe division symbolized by : be carried so far that c is either wholly decimal or partly integral and partly decimal, 7 is a decimal less than unity. The following are instances of this :— 25 0-051 IE al =0:409 + 61 hs pe Ute Slop) 20015 paste IT. 35 = 9:°545 + 33 ee) Oileae 12 5. , 0:006942 . IRLE. 7B = 527 + —agrg =O 006942. * Communicated by the Author, having been read before the British Association, Section A, September 9, 1876. . 3846 Mr.W.H. Walenn on Division-Remainders in Arithmetic. Thus, in the ordinary way of considering the operation of di- vision, not only is it necessary to know “the quotient in order to obtain the remainder, but the value of the remainder is de- pendent upon the 5 alue of the last figure of the quotient. This is shown by the fact that if example I. were carried one step ee the quotient would be 0:4098 and the remainder 0°0022 If, however, in the eqr uation > =c+-—, ¢ be put out of con- b i sideration, if a and b be integral, and if successive numerical values (as 1, 2, 3, &c.) be as ssioned to a, the corresponding values of r sell alee be successive ; they will moreover be periodic, having for their period a value equal to 6. For in- stance, when 0 is realized mentally in reference to any . remainder it may possibly ke the remainder must be one of the numbers or units 1,2, 3, 4, 5, 6, 7, 8, 9, taking, for convenience, 9 to be the remainder when the dividend is an exact multiple of the divisor, so as to include all instances without the use of the symbol 0. This is equally true of the general statement > a ", when 0 is any whole number ; for the point borne in mind in this way of regarding division is the divisor, in reference to whether it is contained a whole number of times or not in the dividend, quite independent of the quotient. If it be contained a whole number of times, the divisor may be put down as the remainder ; if it be not contained a whole number of times, the remainder (in respect of the number of units that make up the divisor) must be one of those units. Considering the remainder to a division in this ight, no remainder can be a fractional quan- tity if the dividend be finite and the divisor integral. Manifestly the use of looking upon the remainder to a divi- sor in relation to a function that contains the quotient, as in the formula 5 Sets i? is either to be able to continue the division to the next figure, or to obtain the result of the divi- sion itself with rigid and absolute accuracy. Viewed in this light, 7 has no other uses and no other properties. If, however, the quantity ¢ be put out of consideration, and the remainder be viewed as being made up of units from unity itself to the divisor itself, both inclusive, the case is quite dif- ferent ; the remainder becomes an entity or function which has properties of its own, that not only apply to the particular - instance whence it has its being, but which may be used ¥ Mr. W. H. Walenn on Division-Remainders in Arithmetic. 347 wherever calculations have value. More than a thousand years ago the Arabians used the tarazu or balance to check the operations of multiplication and division ; the operation called tarazu is that of casting out the nines, or ascertaining the remainder to 9 that any given number has, and with these remainders performing the same operations as with the num- bers whence they are derived. Lucas de Borgo, in the fifteenth century, used the remain- ders to division by 9, as well as to division by 7, obtained for the various data in a numerical calculation (each datum having its corresponding remainder), to check the operations of addi- tion, subtraction, multiplication, and division. This employ- ment of the remainder to a divisor is perhaps the first in- stance of the use of a corresponding or factitious number or function in computations; for logarithms were invented by Baron Napier long after—namely, in 1614. Mathematicians of recent times have extended the isolated observations of Lucas de Borgo, and have proved the still more general principle that all direct and some inverse opera- tions upon the remainders to divisors are respectively analo- gous to the same operations upon the dividends. This now appears evident from the parcelling out of numerical values which division by a constant divisor affords. The difference between these remainders, as factitious numbers or functions, and logarithms, is, that in the first case the same operation— but‘in a shorter and more condensed form—is used to obtain a result ; whereas, in the second case, the factitious numbers are capable of being dealt with by an operation one degree lower in the scale of operations. Moreover, in the case of remainders, the application is possible in consequence of their periodicity, and the original result of the calculation is not obtainable, but is simply analogized ; whereas the logarithms being continuous functions, give results quite parallel to those of the intended calculation, and therefore translatable into the answer by suit- able calculations or references. Looking upon these remainders as functions of numbers, it is important that their use in performing analogous calcula- tions to a primary one should be recognized and made possible by means of Tables, also that a method should be obtained for their easy discovery. From what has been published in the Transactions of the Sections of the British Association for the Advancement of Science (see the ‘ Report’ for 1870), it appears that the nu- merical theorem therein put forth by the author of the present paper gives a means of ascertaining the remainder to a divisor without knowing any thing of the value of the quotient. This 348 Mr. W. H. Walenn on Division-Remainders in Arithmetic. theorem is that (L0O—6)t+u has the same remainder to divi-. sion by 6 that 10¢+w has, 10¢+w being a given number in terms of its digits, and 6 being any whole number ; this for- mula, which is a function of 6, but not of the quotient of the division, may be expanded to the form (10—8)"""'an+ (10—6)"7a,_) +... +(10—6)’a; ot (10—68)ay + Ay. This form is well adapted for use in general calculations. Since a, is the extreme right-hand digit of any number, it is only necessary that the dividend should be finite ; a, may either be the unit digit or a digit on the right hand of the de- cimal point. From this it ‘appears that although the divisor must be integral, the dividend may be fractional. That (10—6)t+wu and 10¢+u have the same remainder to 6 may be seen from the fact, deduced by multiplication, that (10—d)t+u=10t+u—ét. In the former paper the name wnitation is assigned to the above operation upon a given number, which yields the re- mainder or unitate to a chosen divisor or base. The function called the unitate of 2 to the base 6 is symbolized by U,«. Then U;2=(10—8)""'a, + 10-8)" an_y +... + (10—8)?a; | + (10—6)a.+a, if 2=10" a, +10" ana +... +10?a3 + 10a, + a4. The function U,(—~) is easily derived from U,w. For—a may have 6 added to it without altering the value of U,(—z). Hence U,(—2)=U,(6—~2); that is, U,(—~) is the comple- ment of « to 6. For instance, U,(—3)=—U,(9—3)=6. Thus, for every value of U,(—.) there is a corresponding value of U,#, and for every value of U,« there is a corresponding value of U,(—z2). The function U (") may be investigated by means of its equivalent U, (« =U, .y~'). In the case of 6 being a prime number, all the values of this function are integral, ex- : i : cepting when y=0; as Uns =A4, Whe =8. When dis not ie: ; a prime number, U,~ is only integral when U,y is not equal | Y Mr. W. H. Walenn on Division-Remainders in Arithmetic. 349 Loan 6 The laws of reciprocals should be satisfied by unitates; thus to 6, or nota submultiple thereof ; for instance, Uy : == Waox Uy, : should be equal to unity, and accordingly Un(3 x 4)=U_12=1. The function U,«", in the case of 6 being a prime number and n being the variable, recurs in 6—1 terms. When 6 is not prime, ‘the period of recurrence is less than 6—1. When « is the variable and n an even number, it recurs in 6 terms, and 6—1 of those terms are arranged in groups, the latter half of each group being the same as the first half but in inverse order ; thus eee 43. Wr: 9,3, 4,5, 13° 115.1, 5, 4,3, 95 de. The function U,«~” behaves itself like U,«~! in respect of its values being integral or fractional. 1 The function U,«” may sometimes be ascertained in whole numbers from continuing the series U;7”". When 6 is of the 1 form p"—«x (n and & being whole numbers, and 2” or ,vx 1 being irrational), U,«” is finite and integral. In unitation, as in some other numerical operations, there may be m mth roots, n nth roots, and so on. Taking the function Ua as true when «= %7q¢ (an irra- tional quantity), it can be proved to be possible to assign the values of the extreme right-hand figures of some incommen- surable quantities. For instance, UpV5=5, Uyov/29=9, Uyoo4/18=17. The formula U,(—7)=U,(6—w) may be applied to func- tions of ./ —1; thus, U,(a+b/ —c)=U,(at+tb/b—c). In this manner \/ ye in unitates may be expressed asa quantity that is not imaginary, without the use of geometry. The investigation of the function U,« has not proceeded beyond this ne but several aprecions of its properties have been made. Checking tables and calculations that can be finited is among the most practical applications. All divisions to be tested are obliged to be completely finished by obtaining all the figures of the quotient together with the “remainder 5 in Do this case the division of unitates may be performed in the bh : 350 Mr. W. H. Walenn on Division-Remainders in Arithmetic. same manner as ordinary division, reducing the values from time to time. For instance, to obtain 9) 2¢ 2 269 =U, (22 a 122 122 , 200 eO Z Use = Baty 3 act 43x 3E7, 122 5 D and U, (22 y= H44 7 =446%2 =U,16=7. Many theorems that, treated in the ordinary way, involve ee cumbersome Agee of algebra used to deal with scales of otation, can be investigated by the simplest means. For in- ee the proof that the addition of the digits of a number casts out the nines of that number is manifest from the sub- err of 9 for 6 in the general formula of unitation given above; for in that case each term has 1 for its coefieients In the function Uy, the coefiicients are, beginning vas the unit, +1 and —1 alternately. : These relations of U,e (generally written Uz as being the most common of allthe systems of unitation) and of Uyz, give the following singular property, an extension of a well-known fact. If dots or marks be made at equal intervals over the digits of a given number, or part of a given number, and (haying regard to the relative position of the marked digits in the number) if the marked digits be transposed among themselves, the remainder obtained by the subtraction of the transposed number from the original number—or vice versd, according to which number is the greatest—will be divisible by a number composed entirely of ones, as 1111, also by some other number composed entirely of nines, as 9999, the number of ones or of nines in the divisor being a +1, if there be a in- tervals. for instance, I Ili Il 2 Ete: tat 98212381—12385821=4585410, and 4585410 is exactly divisible by 111 and by 999. An easy and practical method of constructing decimal equi- valents for reciprocals is as follows :—Take a ‘reciprocal with any number for its denominator that has 9 for its unit figure, and employ as a multiplier a number greater by 1 than that 7 sees Pi represented by the remaining figure or figures. In 7p 238 the multiplier. Be eginning with 1, this multiplier 1 is used to obtain the next figure towards the left from the previous one as trom a multiplicand, adding-in the tens digit or other digits Mr. W. H. Walenn on Division-Remainders in Arithmetic. 351 that may be carried from the previous product. In this way the decimal equivalent for a is found to be 29 0-03448,27586,20689,65517,24137,931, HS, Pe ea 7 7 = 49 17> 19° &e., this plan can be used to give the figures of reciprocals that have 7 for the right-hand figure of their period ; the operation in this case is begun with 7 instead of 1. By similar means, other decimal forms can be constructed; the forms that can be deci- 1 1 10a+9 10a+7 10a+3’ and Silaia The other decimal forms of reciprocals are either 10a+1 finite, or they may be derived from the form d being the multiplier. Since mally wrought out by this means are J! 10a+9° A method, which is sometimes easier than the above, is to use the multiplier as a divisor, beginning with the last figure of the period, working towards the right hand, and taking each figure of the quotient as it is obtained as the figure to be brought down for the unit figure of the dividend at each ope- ration. The work to obtain “(= in) by this method is :—5 into 7, 1 time and 2 over; 5 into 21, 4 times and 1 over; 9 into 14, 2 times and 4 over; 5 into 42, 8 times and 2 over ; 5 into 28, 5 times and 3 over; 5 into 35, 7 times and 0 over; ae 07,1 time ; and so on, giving the digits (7) 1428571, ue. In reference to unknown quantities, the power of unitation to ascertain the remainders to divisors thereof may be shown by an example. The unknown quantity must be connected with a known quantity by definite laws. If, for instance, it be required to take the 647th power of 256 and to ascertain the remainder, to the divisor 7, of that high number, the fol- lowing is the work, taking into account that the function U,« has 6 for its repeating period :—U,256=4, and U,647=5; piem Up4>—2. Therefore U,256%' —29. The function U;f, f being a figurate number, has, to some extent, been investigated. U,f does not repeat, according to any law that has yet been discovered; but groups of these unitates repeat ; those that have been examined repeat either atter 9 terms or after 9n terms. U,,f being formed by a square arrangement of the numbers as they are obtained, re- peats after 11 terms in the horizontal as well as in the vertical 252 Mr. W. H. Walenn on Division-Remainders in Arithmetic. direction. Taking the first and repre- sentative square (which has only ones for the numbers in its periphery), the 1 Il Libadd diagonal and the triangular half which L29345641 has its apex in the lower right-hand 13643 3 4 corner, its hypotenuse being the dia- T442526 gonal, has 11 for all its contained values. 153946 6 As U,/ behaves itself similarly to U,,/ 16326 6 6 ; ; 114 6°56 76m6 in this respect, it is probable that prime numbers for the value of 6 favour this formation. The unitation square of U,/is in the margin. Of the eight perfect numbers that rare known (calling the serles p), a ‘few of the unitation functions are :— Usp 2 2,2 Up =o) ial alee te Up 2, AA A ae ae Upp me Ep leo lee 3,3 Dp 61h Als AeA oe 6, 6; 1, 10,6 6. Algebraical geometry applied to unitation yields some striking results, especially when polar coordinates are used. The poly ons obtained by thus setting out the values of func- tions of Ux, in reference to the corresponding and consecu- tive whole-number values of Uw itself, represent graphically those functions In a convenient form. The values of functions as given above, also the properties of unitates that are put forth, must be regarded as results given as simple statements ; ‘lve proofs of these and of some of the applications of the general formula, together with Tables and methods of calculating unitates, may be found in a series of articles now publishing it in the Philosophical Magazine. The published papers are dated :—I. November 1868; Il. July 1373; U1, “May 18755 IV. VAmoustimis(on We ‘December (Supplement). 1875; VI. June (Supplement), 1876. Unitates manifest themselves as functions which may be found throughout the whole domain of quantity ; they are in some instances interpretable when the quantity from which they are derived is but imperfectly known. 74 Brecknock Road, N. August, 1876. frs5a° | XLIV. On a Model illustrating Mechanically the Passage of Hlectricity through Metals, Electrolytes, and Dielectrics, ac- cording to Maxwell’s Theory. By OutverR J. Lopas, B.Sc.* [Plate III] Sel: ae obstructions which an electric current meets with in its passage through a medium are of two kinds, viz. opposition electromotive force and resistance proper. The first, since it does not necessarily change sign nor vanish with the current, has a tendency to produce acountercurrent. The second opposes the current with a force which is always pro- portional to the current and always acts against it; it can therefore produce no reverse effects. Since the laws of the flow of electricity are in most respects the same as those of an incompressible fluid moving ina closed circuit, we may represent a current of electricity in a conduct- ing medium by a flexible inextensible endless cord circulating continuously over pulleys. When the cord is at rest in the conducting medium, it is no more (and no less) to be called electricity than an ordinary copper wire is to be said to con- tain electricity at rest; but when it is In motion it is to be called an electric current, and the strength of the current through unit area of the conductor is to be measured by the quantity of cord which passes any fixed point in a second ; in other words, the strength of the current is proportional to the velocity of the cord. The cord can be made to move by ap- plying to any part of it, in the direction of its length, a force which corresponds to electromotive force, and which can be generated in various ways, as by a winch, a weight, or an elastic string. To represent the resistance of conductors, the cord may be made to rub against rough surfaces ; and to represent opposi- tion electromotive force, the motion of the cord may be made to stretch or to squeeze elastic bodies, thereby setting up a ten- sion in them which shall not only oppose the motion, but shall also have a tendency to force the cord in the reverse direction. The difference of potential between any two points of a cir- cuit is represented by the difference between the tensions of the cord at the two points ; a greater tension is to represent negative potential, while a less tension or a pressure is to re- present positive potential. If the cord passes through a body, and if by any means we can make that body inciude more of the cord than its normal allowance, then that body is said to be positively charged with * Communicated by the Author, having been read before Section A of the British Association in Glasgow, September 1876. Pie vag. >. 9. Vol. 2. No. 12. Nov. 1876. 2 A B54 Mr, OF9;: Lodge on a Mechanical Lllustration electricity ; if, on the other hand, we can abstract some of the cord from the body, it is negatively charged. The capacity of the body will be measured in the ordinary way by the excess of cord which is got into or out of the body, divided by the difference of potential required to get it in or out. Model to illustrate the properties of Dielectrics. § 2. Now let us apply these considerations to the construction of a model which shall represent the charging of a Leyden jar. In fig. 1 (Plate III.) A BC D represents the cord, passing over four pulleys, A, B, C, D,and through eight buttons, 1, 2, 3, &c., which represent eight strata in the dielectric of a Leyden jar or other accumulator ; these buttons are supported on elastic strings, and they grip the cord tightly so as only to allow it to slip through them very slowly and with considerable resist- ance, if at all. Continuous electromotive force is represented as being applied to the cord at D, in the shape of a weight W, tending to turn one of the pulleys on which the cord runs ; and there is a contact-breaker at C, represented by a screw 8, which is able to clamp the cord, or in other words to make the resistance of the external cireuit infinite. The inner coat- ing of the jar may be represented by the space between A and the first button ; the outer coating by-the space between the last button and B. The part ABCD of the circuit, since it contains ordinarily next to no resistance, represents very good metallic conduction. Suppose initially that the cord is free at C, and that the weight W has never been applied ; all the elastics will be ver- tical, the buttons will be at the same distance from one another, the distance A 1 equal to the distance B 8, and there will be no excess of tension in any part of the cord. This represents an uncharged Leyden jar, its inner and outer coating connected by a metallic wire. : Introduce into the circuit at D an electromotor (like W). The potential towards A rises, the potential towards C falls, and the cord begins to move from A to B. The first effect of this motion is to displace the buttons all the same distance to the right—to such a distance, namely, that the sum of the ten- sions in their elastics is equal to the weight W. Then the cord stops. The dielectric is polarized; and its condition is shown in I, fig. 2. Electricity has been displaced in it in the direction of the electromotive force: the inner coating A 1 has gained cord; it is charged positively; the outer coating B 8 is charged negatively ; pressure is exerted on the left face of every button, and tension on its right face ; that is, every stratum is electrified positively on the side from which the — of Electric Induction and Conduction. aay ay5: electricity came, negatively on the other side. The potential gradually falls from A to B, 7. e. the tension in the cord gra- dually increases as we pass along it from A towards B. The dielectric is in a state of strain represented by the stretching of the elastics ; and it is tending to produce a countercurrent, 1. €. to drive the cord back again. § 3. Now clamp the cord by means of the screw §, take off W, and leave the whole to itself for some time. Several things may happen. 3 Ist. The buttons may remain exactly where they were, which represents a dielectric of perfect insulating power : the jar will in this case remain charged for any length of time, and the whole charge can be got out of it at any time by unclamping the cord. In the symbols of § 8, d d To) pi), os = a St Se pa =O). As soon as the screw 8 is unclamped all the buttons spring back, overshoot their mean position, and execute a rapid series of decreasing oscillations, carrying the cord with them back- wards and forwards until they come to rest. The discharge of a jar, then, is not a simple passage of electricity in one direc- tion only, but is a rapid series of discharges in alternately op- posite directions, which last until the energy stored up as strain in the dielectric is all converted into heat by the resistance of the circuit. ‘This succession of alternately opposite sparks Fedder- sen” has observed experimentally by looking at the apparently simple spark of a Leyden jar in a revolving mirror. If it were possible to close the circuit so rapidly that very little of the discharge should have taken place before good metallic contact was made, the succession of opposite currents would probably last quite an appreciable time. They might perhaps be detected by discharging the jar through a galvanometer and an electrodynamometer at the same time ; the relation between their indications would probably be different in this case from what it is when either a continuous current or a simple one- directioned discharge is sent through them both. 2ndly. The buttons may all slowly and equally slide back towards their normal position. This will happen if the relation between their friction on the cord and the elasticity of their strings is the same for all. The model will in this case repre- sent a homogeneous dielectric with feeble conducting-power. * Pogg. Ann. vol. cili. Feddersen saw the spark drawn out into a suc- cession of bright parallel lines at continually increasing intervals. Max- well’s theory, equation (2), § 9, appears to make the intervals all equal, but only on the erroneous assumption that the circuit is devoid of resistance. 2X, 2 356 Mr. O. J. Lodge on a Mechanical Illustration If left to itself long enough, such a jar would be found com- pletely discharged ; and since all the buttons would have pre- served their original distance from one another, there would be no internal charge in any part of the jar; ana residual charge is impossible. The values of the symbols of § 8 for this case are ig, bs yee no ne i Sjo= -a- o> Lissa eens eo el anna oe and after charging it by means of an electromotive force HE, and leaving it for ¢ any time ¢, the whole charge can be got out of it at once, and its amount ‘will be t CHi,e ©R. 3rdly. Some of the buttons may be smoother or may have stronger elastics than others; and in such case they will slide back more quickly than the others and will represent strata of comparatively high conducting-power. Thus in fig. 1, or in II (fig. 2), 38, 5, and 6 are such buttons, especially 5 ; while 1,4, 7, and 8 have remained pretty steady without slipping. What, then, is the condition of the dielectric in this state? The slipping back of some of the buttons will have relieved the ten- sion on the cord, in other words, will have diminished the differ- ence of potential of the two coatings ; ; but no motion of the cord has taken place either way because of the clamp atC. In the figure the buttons i and 8 have remained tight on the cord; consequently the charges on the two coatings happen to have remained constant ; but charges have appeared in between the strata. For instance: between 3 and 4 there is too much cord; the dielectric is there positively charged ; between 4 and 5 it is negatively charged. ‘The difference of potential between the two faces of the stratum 5 has almost vanished. § 4. Now unclamp the screw §, 7. e. make metallic contact between the two coatings: a rush of electricity takes place; the buttons spring back, and after a few oscillations take up the position of equilibrium shown in ITi, fig. 2. They have in fact been all displaced the same distance to the left; and so those which (like 3, 5, and 6) had, before discharge, slid almost to their mean position are now displaced beyond it, and the strain in their elastics 1s acting in the reverse direction to what it did before ; consequently the elastics of 1, 2, 4, 7, and 8 have not been able quite to resume their normal length, but they remain in such a state of strain as to exert as much stress one way as the elastics of 3, 5, and 6 are exerting in the other direction. The jar then appears discharged, but we have not yet got the whole charge out of it. Screw down § again and wait. of Electric Induction and Conduction. By) (5 The slippery buttons 3, 5, and 6 will again gradually slide back almost to their mean ‘position, the tension which their elas- ties exerted towards the right will be relieved, and we shall pre- sently have an unbalanced tension in the same direction as at first: this is the “ residual charge;” and the state of things is re- presented in III, fig.2. Onunscrewing Sasecond motion of the cord will take place, and the buttons will take up a fresh posi- sition of temporary equilibrium. ‘The numbers annexed to the four eases I, IT, LEI, and IIT indicate the electromotive force, or the stress on the cord, in each case. Thus when first charged it is 24 ; aiter standing some time it falls (say) to 17; immediately after the first discharge i itis of course 0; and after again waiting some time itrises to 3 in the same direction as before, again to fall to zero when the second discharge occurs. Still, ‘how rever, the jar is not wholly discharged, because the slippery buttons will again have been displaced beyond their mean position and will slide back again ; so it will take an infinite number of mo- mentary contacts completely to discharge the jar, unless indeed every stratum allows a little slipping to go on. This is Professor Clerk Maxweil’s theory of a composite dielectric, which he explains in art. 328 of his ‘ Blect: icity and Magnetism,’ and which he also illustrates mechanically in art. 334. § 5. If any button is perfectly smooth, it does not get dis- placed at all, and it therefore takes no part i in the action. It would represent a film of gold leaf lying inside the glass of the jar parallel with the coatings ; and it would become oppo- sitely charged with electricity on its two sides—negative on the side facing the positively charged coating, and positive on the other ; for the other buttons would evidently move closer up to it on one side, and away from it on the other. This, together with the preceding, really illustrates the whole of the phenomena ordinarily spoken of as charge in conductors according to the views of Faraday and Maxwell; for to elec- trify any conductor by induction, you place it inside a polar-. ized dielectric (usually the air), and the opposite electricities appear on its surface exactly as in the above piece of gold leaf; while to ‘ “charge a conductor”? with one kind of electricity ou use the conductor to coat one face ofa polarized dielectric, which is done by judiciously relieving its state of strain on one side only of the conductor you wish to charge. Conduc- tors then are to be looked upon statically as mere interruptors in the continuity of a dielectric medium; and if the medium is polarized, charge appears at the surface of discontinuity. Regarded in this way, the analogy between electrification and magnetization is very close; and we can no more get positive 358 Mr. O. J. Lodge on a Mechanical Illustration electricity detached from an equal quantity of negative than we can get a north pole detached from a south: only in one case steel is the polarized medium, in the other case air. § 6. Instead of making momentary contacts as in § 4, we might leave the screw Sup so as to obtain a feeble continuous current from the dielectric by reason of the gradual slipping of some of its buttons, till it is completely discharged. The strength of this current at any instant Maxwell calculates in art. 330. Again, after having charged the dielectric, instead of re- moving W, we might leave it on and not clamp the cord. This would represent a jar permanently connected with an electromotor maintaining a constant difference of potential. If none of the buttons slipped on the cord, nothing would in this case happen after the first charge. If all the buttons slipped equally, we should geta feeble continuous conduction- current through the dielectric after the first rush due to elec- trical displacement had subsided. But if some of the buttons remained firm on the cord while others slipped, we should have a slight continuous current not purely of conduction, nor purely of displacement, but of both combined. There would be conduction through the slipping buttons, and there would be displacement of the tight ones with increasing tension in their elastics to make up for the relaxation of the others ; and this might go on for a long time, the current continually getting weaker. This is actually observed in the dielectric of submarine cables (cf. Maxwell, art. 866). The insulating material of a cable, in fact, consists of layers of different mate- rials (gutta percha, Chatterton’s compound, &c.) laid in strata over the conductor, as if these residual-charge effects were exactly what was wanted to make a good cable. I suppose that they do not exert any detrimental influence on the use of a cable, the method of signalling being what it now is; but if it were desired to lessen these effects, it might be done by rendering the dielectric as homogeneous as possible, first of all by making i of only one material, and secondly by laying this material on the conductor by some process which shall not necessarily develop in it a stratified structure as the present method does. Disruptive Discharge. § 7. When the electromotor is acting on the dielectric in the way last described, the strain in some of the elastics may become so great as to break some of them ; we should then have partial or complete disruptive discharge, and the subse- quent resistance of the dielectric would be much diminished. of Electric Induction and Conduction. 359 Such disruptive discharge as this represented by a snapping of the elastics is analogous to the rupture of solid dielectrics hke glass. In fluid dielectrics like air, the disruptive discharge may be better re epresented by a slipping of the pins to which the elastics are fastened along a groove parallel with the cord: at each slip the cord makes a bound forward; but no perma- nent damage is done. (If at any one point a slip takes place, the extra stress thrown on to the others is liable to make them give way too.) A discharge of this sort is a kind of electro- lysis, and may be supposed - to go on slowly in glass at a high temperature. it would be accompanied by a polarization of the electrodes, as we shall see later (§§ 13 and 15). Discharge ina vacuum-tube seems to be of this nature. If partial disruptive discharge occurs at points inside a dielectric, it will in general give rise to interna! charge, even though the dielectric be of uniform conductivity, because the buttons will not preserve their original distances from one another on the cord. Maxwell’s Theory of a Composite Dielectric. § 8. Let us now see how the model must be made in order to agree accurately with the dielectric, as investigated in art. 328, vol. i. of Maxwell’s ‘ Beomel It is there supposed, “for the sake of simplicity, that the dielectric consists of a number of plane strata of different ma- _ terials and of area unity, and that the electric force acts in the direction of the normal to the strata.” The row of buttons in our model (fig. 1) will represent this collection of strata; and we shall, for simplicity, consider the normal distance be- tween adjacent buttons in the row as unity. The thickness q, ,.-.. Of any stratum is then equal to the number of buttons (or molecules) which compose it; and the whole thickness of the dielectric a,+a,+....+¢» will equal n, the whole number of buttons. Since all the buttons in any one stratum have exactly the same properties, we may if we like consider each button to represent an entire str atum, and ‘take a,=a,=...=1: and this is what we shall virtually do in what follows. The other symbols employed in the investigation are the following, with their meanings affixed (1st, the electrical one after Maxwell, and 2nd, the corresponding meaning expressed in terms of the mechanical model) :— X, Xz, &e. [the resultant electrical force within each stra- tum]. The tension in the elastic of each button if the buttons are unit distance apart, or, more generally, the difference between the pressures on the corresponding faces of two consecutive buttons divided by the distance between them. 360 Mr. O. J. Lodge on a Mechanical Illustration Pis Pa, &e. [the current due to conduction through each stratum]. The rate at which the cord slips through each button. thy fo, ke. [the electric displacement]. The distance of each button * from its mean position. Wy, U2, &e. [the total current through each stratum, due partly to conduction and partly to variation of displace- ment]. The rate at which the cord passes by a point in each stratum fixed in space. 715 1, Kc. [ the specific resistance referred to unit of volume |. The coetticient of “ friction” (see § 10) between the cord and each button. K,, Ko, &e. [the coefficient of electric elasticity, which is equal to = where K,, K,, &c. are the specific inductive capacities of the several strata (see Maxwell, art. 60) ]. The “ coefiicient of elasticity ”’ of each elastic, or the force tending to replace a button when displaced unit distance from its mean position (see § 10). | E [the electromotive force due to a voltaic battery placed in any part of the external circuit]. The force applied at an external point such as D (fig. 1) to move the cord, or the tension in the part of the cord between D and C minus that in the part between D and A. u [the current in the external circuit and battery]. The rate at which the cord passes over the fixed pulleys. Q the total quantity of ee ‘ which has passed t through the external circuit up to the time ¢. Q=( udt. 0 R, [the resistance of the battery with connecting wires]. The drag on the cord in the external circuit, including elec- tromotor at D, pulleys, and any other rubbing surfaces. as S. * In the model, then, electric displacement in a dielectric is accompa- nied by displacement of matter. It may possibly be so also im fact; and if so, on suddenly charging or discharging a condenser, a sudden kick might be observed, the body as a whole being thrown forward with a mo- mentum equal and opposite to that of its displaced particles. But thisis not at all necessary, because one does not see to what the other ends of the elastics can be attached except to another set of buttons which get dis- placed in the opposite direction by a negative current simultaneous with, and equal to, the recognized positive current. (This set of buttons must be understood to take the place of the fixed beams employed in the model for convenience.) As, however, there is no reason apparent why the masses of the two sets should be equal, a kick might be observed equal to the difference of the two momenta; at any rate it is worth looking for. of Electric Induction and Conduction. — 361 ‘Oy. [the surface-density of electricity on the surface which separates the first and second strata]. The length of cord in excess of the normal length, which exists between the first and second buttons. R the regular Ohmic resistance of the dielectric, or . AN + glo + eee. t+ An?n- C the capacity of the dielectric measured instantaneously, 2. e. without allowing time for any slipping between cord dOnt and buttons, or IE? it equals eee aie § 9. Maxwell then goes on to consider what happens in any one stratum (say, the first) when the electric forces are brought to bear on the dielectric. By Ohm’s law we must have Oy Xy = A471 Pi. ~ C C 5 c - z, (1) Maxwell’s theory assumes MG = Ky fy 3. sects ° . ° . . (2) and by definition of total current, d Uy — i + os . ° : ° A 6 . (3) Also evidently . 079 =fo—hfi, ° ° . ° . . . (4) do15 Sa =P1— P2- : . C A - 3 (5) and These equations being given, certain consequences follow, of which the most obyious are Uy Ug Un=Uu, . . . . (0) and H=a,X,+a,X_4+...., ° 5 4 = (10) while the most important is a relation between E and wu, which is obtained by substituting in (3) the values of p and / from (1) and (2), and then adding up in accordance with (10) thus, H=(a,D;+a,D,+....)u,.. » ». Al) : — where D, stands for the operator é + st 5) : il Ky dt Equations (4) and (2) give us which shows that internal charges will appear unless the stretch- 362 Mr. O. J. Lodge on a Mechanical Illustration ing forces are proportional to the elasticities of the several threads, or unless these elasticities are infinite (as they practically are in the case of metals). If an accumulator be charged by means of an electromotive foree EH, continued so long that the p become uniform and constant and equal to By the last equation may be written, by help of (1), R’ O79 =(2 = Ale ° . . . . (19) which proves that there will be no internal charge even in this case if the v are proportional to the « (cf. case 2, § 3). After having charged the jar with this long-continued force ij,, discharge it instantaneously, and then leave it to itself for atime ¢ with S screwed down. ‘The difference of potential between the two coatings, which was at first zero, rises by irre- gular slipping of the buttons; and its value EH at the end of any time is given by (24), which we may write KH Ary 1, Ky = ly Ag?9 AgKy, eae = t Ny (Stas) > (ak) ee Ca stay) iin A quantity of electricity HC can at the same time be got out of the jar; and this is the residual charge. Conditions required of the Model in order to be an accurate representation. § 10. The conditions which the model does not naturally satisfy are (1) and (2). Condition (2), however, that the re- storing force shall vary with the displacement, is nearly satis- fied by fig. 1 for small displacements ; and it can be made accurately true by passing each elastic through a smooth fixed ring placed in the line of its mean position at a distance from the fixed pin equal to the natural length of the unstretched elastic. Then (2) is satisfied by the model; and each button, if pulled aside, would execute a succession of simple harmonic motions about its mean position. To satisfy condition (1) is not quite so easy. Electrical resistance is very different from ordinary friction, though they are to some extent analogous. The smallest electromotive force is sufficient to produce some current against any resist- ance however great, and the current increases proportionally to the electromotive force from zero upwards ; but friction is a discontinuous quantity. For small values of force applied to move a body, the friction is exactly equal to the force, and the body does not move ; but as the force increases, at a cer- tain critical point the friction suddenly falls from the value (24) of Electric Induction and Conduction. 363 of the force at that instant to something less, and at this value it continues constant as long as the body continues to move. ‘This appears nearly what is wanted to imitate disrup- tive discharge in a vacuum-tube (see Mr. Varley’s experi- ments, Maxwell, art. 369); but in the case of a metallic con- ductor or an ordinary dielectric, the cord is required to slip through the buttons, if it slip at all, even when the force is slight ; and the rate of slip is always to be proportional to the force. This might be roughly attained in the model by making each button of a short piece of india-rubber tube well greased inside, and by attaching the ends of its elastics to a ring fixed on the tube near one end, so that when it was displaced the pull on its elastics should make it grip the cord more tightly, and thus keep the ratio, between the force pulling the cord through and the rate at which it came through, roughly con- stant. But it would be probably unnecessary to do this ; for by making very smooth and well-oiled buttons pinch the cord tightly, one could arrange that the cord should be able to ooze slowly through them without sticking; and then the residual- charge phenomena would be imitated by the model with suffi- cient closeness. Moreover | have reason to think that friction between cord and metal is not independent of velocity. Metallic Conduction. § 11. In order to make the model represent the flow of electricity through a metallic conductor, we have only to make the resistances 7 small and the elasticities « very great. We thus approximate to a set of smooth buttons supported on rigid fixed rods, as shown in fig. 8. In this case the displace- ments are zero, the whole current is a conduction-current ; and there is no tendency to a reversal of the current by reason of an opposition electromotive force. In symbols, a pe, or —O> re ee Mahe To make it exhibit electrolytic conduction (or rather convec- tion), we shall see in § 13 that the rods supporting the buttons must still be rigid, but that the pins by which they are at- tached to the beams must slide in a long groove parallel with the cord. | Submarine Cables. (See also § 6.) § 12. Ifa circuit consists of a simple wire hung in space, the passage of electricity through it will be represented in the model by a single endless cord; and the cord may be abso- lutely inextensible, so that the propagation of potential takes place instantaneously ; or possibly it may be more in ac- cordance with Maxwell’s theory to say that the cord is so very 364 Mr. O. J. Lodge on a Mechanical Illustration nearly inextensible that impulses are propagated along it with the velocity of light. But if the wire, instead of hanging in free space, is coated with some insulating material and im- bedded in the ground, the propagation of potential goes on comparatively slowly.. There are, in fact, besides the simple circuit straight through the wire and back through the earth,a multitude of lateral circuits, each represented by a separate cord, one for each unit area of the dielectric. When an electromotive force, say, a negative potential ora pull, acts at one end of sucha compound circuit, its first effect is to pull out those cords which offer least resistance ; in fact’ the current is split among the several circuits in the inverse ratio of their respective resist- ances. The direct and longest cord feels a slight instanta- neous pull indeed; but it is only slight until the cords through the dielectric have been pulled and a tension set up all along the line. While this is going on, the pull on the direct cord is gradually increasing. It would be possible to represent all this by a sufficiently elaborate model ; but the action may be imitated with only a single cord if it be arranged zigzag on pulleys supported by elastic strings, as shown in fig. 4. A sudden puli at one end of such a cord will stretch the elastics of the near pulleys ; the strain so caused will stretch the next, and so on; and thus the impulse wil be transmitted along the cord, getting, however, weaker and weaker as itadvances. A still simpler imitation of the effects observed would be obtained by making the direct cord itself elastic ; but such an illustration would have very little analogy with the process really going on. Electrolytic Conduction. § 13. A model intended to represent the passage of a cur- rent through an electrolyte is shown in fig. 5 ; but it is not intended to be actually constructed *.. In this figure the buttons are supported by rigid rods; but instead of these being screwed to beams as in fig. 3, they are attached to rings which slide freely on two glass rods a and b. ‘The additional piece Z is a knife-edge mounted on elastic pillars or spiral steel springs, and placed close to the cord, so that its sharp edge is ~ in the path of the buttons if they travel along. The. friction between the cord and the buttons is supposed to be practically infinite. Now hang a weight W on to its hook at D and note what occurs. The cord moves from A toward B, carrying with it the buttons, which travel bodily forward without bending their * The electrolytic model here described is capable of considerable sim- plification and improyement, of Electric Induction and Conduction. -— 365 supports, by reason of the slipping of their rings on a and 6. This represents that no opposition electromotive force is gene- rated in the electrolyte itself, but that it offers a little ordinary resistance to the passage of the current; also that the current is conveyed by a bodily transfer of material—that i1 is, in fact, a convection-current. “Further, since the buttons do “not slip on the cord, the length of cord between any two of them re- mains constant ; 7. e. there is no internal charge developed in the electrolyte. This natural length of cord between two adja- cent buttons may be called the “ molecular charge ;’’ it appears to be the same (up to a simple factor) for all substances, and is what Maxweil calis one molecule of electricity. Now let button No. 9 reach the electrode Z; it travels on, compressing the elastic supports and calling out an opposition force which may be sufficient to check the ‘descent of W, or it may not. Suppose that it is not; then the button still moves on, its pressure against the knife-edge increases, until the button is ent in half and released from the cord. The tension being thus suddenly relieved, the cord makes a bound forward, the “molecular charge” of No. 9 passes over the pulleys, and No. 8 comes into contact with the electrode, to be opposed, cut, _ and released in like manner. When a button has been cut, it merely hangs by its rings to the rods a and 0, offering a little passive obstruction | because it has to be pushed forward by the others ; but it presently gets pushed off the ends of the rods, and it then falls away. ‘The cord thus progresses In a ra pid succession of jerks ; ‘but as in any actual circuit the number of cords parallel to one another is very great, the average motion is continuous and not jerky. The procession of buttons towards Z represents the proces- sion of the atoms of the cathion (say by Gregor) toward the cathode ; when ey reach it, they at first develop an opposi- tion electromotive force at its surface : ; but presently getting released, they deliver up their molecular charges to it, rise through the liquid, slightly increasing its resistance, and escape as gas. In order to diminish the polarization of the electrode, the knife-edge may be sharpened, which corresponds to making the electrode of gold or platinum instead of copper or iron ; or the buttons may be softened and rendered easier to cut, which represents either a cathion less electro-positive than hydrogen, or else the addition of a solvent like nitric acid. Strengthen- ing or weakening the springs which support Z does not affect the electromotive force called out; but it does alter the time during which the reverse current lasts. Platinum (when not covered with “black,” asin Smee’s cell) must be 3566 Mr. O. J. Lodge on a Mechanical Illustration supposed to have weak springs; for the hydrogen clings tightly to it, and the reverse current, occasioned by the expansion of the compressed springs, lasts a considerable time. I have supposed the buttons rigidly stuck to the cord ; but if any electrolyte allows a trace of electricity to pass through it by ordinary conduction without equivalent electrolysis, then the cord must be allowed to slip through the buttons a little*. § 14. But the model is as yet only half a one; it does not represent the oxygen proceeding up stream to the anode. To show this we require another set of buttons, ranged alongside the first set, on a cord which is driven inthe reverse direction by the same weight W, and which represents a simultaneous equal opposite current of negative electricity. These buttons are also attached to rings sliding on rods a’ and b’; and these rods a’ and b’ must le close to a and 0 respectively, so that the H buttons and the O buttons may rub against one another as they travel in opposite directions. Indeed I imagine that the rods themselves are to be infinitely smooth (and might really be dispensed with, rings and all), so that there is no real metallic resistance in an electrolyte, but only the resistance (which, however, has been found to follow Ohm’s law) due to the clinging of the atoms. This clinging of the atoms is not great—it does not represent chemical combina- tion ; for, according to the theory of Clausius, it is not the chemically combined atoms which are undergoing electrolysis, but some comparatively free ones knocking about among them. Our model then only represents the Clausian atoms. The electrode at which the oxygen atoms are liberated must not be supposed to impede their progress, but on the con- trary to assist it, especially if made of zinc or some metal oc- cupying a similar position in the voltaic series. It may be conceived of as attracting each button, as it comes near, against its knife-edge and cutting it off the cord; it is thus generating electromotive force in the same direction as W, and rendering any external electromotive force unneces- sary for the propulsion of the cord, -provided that the opposi- tion at the cathode does not overpower it. We have here imagined a voltaic cell carrying on electrolysis and genera- ting a current on its own account. ‘The buttons cut off at the anode stick to it by reason of its attraction, and may be con- sidered as combining with a portion of it. They will then tend to block up the circuit unless washed off; and so the pole gets eaten away. Seeing that the smallest force is sufficient to make the move- * Faraday is said to have found this to occur in certain cases. (See Thomson on Electrolysis in the Phil. Mag. for December 1851, p. 481.) of Electric Induction and Conduction. 367. ment of the free atoms preponderate in one direction, it seems worth while to try if incipient decomposition cannot be pro- duced in a solution of double tartrate of iron, or other conve- vient salt, by magnetism. One might use iron electrodes, care- fully depolarized, connected with a galvanometer, and arranged so that either can be magnetized at pleasure. The current produced, if too feeble to be directly observed, might be detected by using it to charge a condenser. But a more likely way would be to send a current through the solution by a very small external electromotive force, first in one direction and then in the other, and to measure it each time ; in one case iron would have to be carried against magnetic force, in the other case with it (cf. Maxwell, art. 263). If the end of a mag- net were cup-shaped and were filled with a magnetolyte, I sup- pose that a continuous action would go on, tending to deepen the hollow, because the middle of a magnet is magnetized more feebly than the external portions. § 15. If we make a heterogeneous electrolyte by taking a number of strata of different materials, such a composite struc- ture will not be capable of internal or of residual charge as a composite dielectric is ; for the buttons remain at their proper distance from one another on the cord, and this distance is the same for all the different substances by Faraday’s law. It is, however, possible for a body to possess the properties both of an electrolyte and of a dielectric; e. g. warm glass *. The flow of a current through such a substance would be imi- tated by making the connexions between the buttons and rings in fig. 5 elastic strings instead of rigid rods. Internal charge &c. would then be possible—not, indeed, as in an ordinary dielectric, by slpping of the buttons on the cord, but by unequal slipping of the rings on the rods. It may be that conduction in dielectrics is always of this electrolytic nature, and not metallic, as was imagined in fig. 1, § 2, &e. Electrolytic Momentum. § 16. The mass of oxygen carried through the liquid in one direction is eight times that of the hydrogen carried in the same time in the opposite direction ; hence, so long as these materials are being accelerated, it seems as if a momentum should be imparted to the liquid as a whole, equal to the differ- ence of momenta generated in the oxygen and hydrogen in the same time. The acceleration lasts only the inappreciably short time during which the current is assuming its constant value. Let v be the velocity imparted to a mass m of hydro- * See Buff (Ann. der Chemie und Pharm.), quoted by Maxwell, art. 271 368 Mr. O. J. Lodge on a Mechanical Illustration gen in this time ; then, if M is the mass of the liquid and con- taining vessel, the vessel will apparently be kicked backwards with an initial velocity v’, where Mo! + mu—8mv=0.- ... | 2 ee Now suppose the containing vessel to be a cylinder of which the electrodes form the ends, each of area a, at a distance apart equal to d; let p be the density of the electrolytic liquid ; and let o” be the maximum density of the electrolyzable hydrogen in the liquid—that is, the mass, per cubic centimetre of the liquid, of hydrogen actually in motion near the cathode under the in- fluence of the current. Then, if h cubic centimetres of hydro- gen gas at the density o be liberated per second by the current when constant, the mass of this hydrogen will be he=ave'.g. © 1. OS ey, The average density of the electrolyzable hydrogen may be con- sidered as 40’; so the whole mass of hydrogen set in motion through the liquid is , m=tado'= ae oe ayy = while the mass of liquid and containing vessel M=adp +p. Hence, neglecting the mass yu of the containing vessel, pa COLT NG; v= M © Dayo) . 5 ° ° . (4) This velocity would be extremely small; but it may be worth looking for. The electrolytic cell should have the plates smail, because their area occurs in the denominator of (4), and it should be suspended at the end of a short counterpoised tor- sion-arm. Any twisting of the suspending thread might be observed with a mirror and scale. The current should not be allowed to last long enough to evolve any gas; or the reaction of the rising gas might swamp the effect looked for. ‘The best plan would seem to be to reverse the current at regular short periods corresponding with the previously ascertained period of oscillation of the torsion-arm. The impulses would then accumulate, and the gas would not form on the plates in any quantity. The connexion indicated by (2) between the velocity of elec- tricity and the mass of moving electrolytic hydrogen in a cubic centimetre of the electrolyte seems to be of interest, of Electric Induction and Conduction. 369 if it be correct, viz. Owe This velocity v is not the rate of propagation of potential ; is the actual velocity of the areas itself (on the hypothesis that it travels with the ions) when passing through that par- ticular cell and liberating ho grammes of hydrogen gas every second. I see no way, however, of ascertaining the unknown density o’, except through a determination of v by an experi- ment founded on equation (4). ; I have hitherto supposed water to be the electrolyte ; but practically, of course, dilute sulphuric acid would be us ed, and it would be the group of atoms SO, which would travel against the current, and not oxygen. Thi is group being forty- eight times as heavy as hydrogen, we must read 47 for 7 = the above formula. It therefore seems reasonable to vee that if the method succeeded and gave measurable results, in- formation might be obtained by means of it concerning the actual chemical action caused directly by the electric current, independent of any secondary chemical action to which the products ultimately liberated at the electrodes may be due. Besides the sudden kick mentioned above, a continuous mo- tion of the vessel seems possible, because fresh matter is con- tinually being set in motion, while the matter already in motion is being ejected. The momentum generated per second from this cause should be equalto47hov. But then the impact of the matter liberated at the electrodes might exert force on them, which would be almost exactly equal and opposite to that exerted on the ions. Various ways, however, suggest themselves of looking for the two forces separately, or of making them assist instead of oppose each other ; and it is easy to arrange so that there shall be no evolution of gas. The impact of the cathion would be advantageously observed by depositing silver on a flat electrode, varnished on one side, and hung to the end.of a torsion-arm—remembering that ro- tation due to the earth’s magnetism, and probably other dis- turbing causes, such as ‘‘ migration of the ions,” will have to be eliminated. Different Electromotors. § 17. We have hitherto supposed the electromotive force to be applied to the cord in the shape of a weight hung to it. This may be said to represent roughly abattery or thermopile * Jt would be more correct to write o’ G — a instead of o' inthis and od equation (2). The whole of § 16, however, is very questionable at present. inimvtage sd. Vol. 2. No, 12. Nov. 1876. 2B 370 Mr. O. J. Lodge on a Mechanical [Illustration exerting a constant electromotive force. But we also, in i consideration of electrolysis, saw a little further into the ‘eal action going on in a voltaic cell ; and we shall probably, na subsequent communication, try to look into the internal mechanism of a thermoelectric joint. There are other ways, howey er, of moving the cord than by a weight. We may ¢ attach a piece of elastic to it, stretch the elastic, and fasten its free end toa nail; the elaciie would then tend to unstretch itself ; and it would pull the cord a certain distance, quickly or slowly according to the resistance of the circuit. This represents a current due to the subsidence of a given difference of potential such as would be produced in a wire by arranging it between a sphere charged with a certain quantity of electricity and the earth, or by using it to jo up the coatings of a charged Leyden jar. A third way of moving the cord would be to wind it round an axle and then to turn the axle with a winch. This would represent electromotive force generating difference of potential or doing other work at a limited rate, as in a magnetoelectric or an ordinary electric machine. To represent the latter, the axle must be thin and the handle turned slowly ; the current produced would then be very weak at the best of times, but it would be little affected by increasing the external resistance, and it would be able to break through the elastics of a di- electric, i.e. to produce a disruptive discharge. If slipping Bic: place between the cord and the driving-apparatus, it means that energy is being wasted by some means, such as friction in a machine or ‘local action” in a battery. Effects of rapidly reversed small Klectromotive Forces. § 18. Instead of pulling the cord continuously in one direc- tion or the other, let us shake it rapidly to and fro. Ina per- fect dielectric no energy will be lost; for the buttons are car- ried with the cord, and their elastics exert a restoring force, which tends of itself to make the cord oscillate with a succes- sion of simple harmonic motions. But in a metallic conductor the cord simply slips through the buttons, no restoring force is called out, and the energy of the displacement is lost by friction. These small to-and-fro motions represent the effect of waves of light falling on the body ; for, whatever the cord does or does not correspond to in nature, Hf certa ainly represents a con- tinuous incompressible medium connecting together the par- ticles of bodies; and supposing light to be small and rapid oscillations of such a medium, it would be represented by small and rapid oscillations of the cord. But the linear extension of of Electric Induction and Conduction. vo Sah a cord, though well enough fitted to picture a linear electric current or a stream-line, is not well adapted for the represen- tation of the continuous medium at rest. One can, however, partly get over this by supposing that the cords are lying in all directions, and that all are connected together in some way*. Let us then picture to ourselves a dielectric medium as an assemblage of particles (or buttons) joined together by elastic strings and threaded by straight inextensible cords which form three mutually connected sets, each set at right angles to the other two. Imagine a set of transverse waves running down the set of cords parallel to the axisofz. The cords in the direction azand y will be shaken backwards and forwards with their but- tons; but no energy will be lost, and the disturbance will pass right through the medium at a rate dependent on the z cords themselves, on the masses of the particles, and on the elasti- city of their joining threads. An insulating medium is then a transparent medium, provided it is sufficiently homogeneous not to scatter the light, and provided it does not contain pig- ments or other foreign absorbent materials. But now form an image of a metallic conducting medium : the cords, as before, thread particles in three cardinal direc- tions ; but the particles are now very smooth and are connected to each other, not by elastic strings, but by nearly rigid rods. Oscillations travelling down the axis of z displace, indeed, the z and y cords; but since these slide through their buttons, no restoring force is thereby called out, and the disturbance ‘does not succeed in penetrating far into the medium before its energy is all lost in friction—converted into heat. A con- ducting body is necessarily an opaque body (see Maxwell, art. 798). In a dielectric the connexion between a button and the cord is good, but the connexion between one button and another is lax. Motion of the cord is therefore readily and quickly im- parted to the buttons, or vice versa; 1. e. dielectrics are good absorbers and radiatorst. Motion of a button is only slowly transmitted to other buttons ; they are in general bad conduc- tors of heat. In a metal, on the other hand, it is not easy to set the buttons swinging by means of the cord ; but ifa button be once set in motion, its motion is rapidly transmitted (by conduction) through the mass. There is here no explanation * Maxwell’s theory is not responsible for the tendency in the model to make out that an electric current is a current of ether. + Cf. Rankine “ On the Hypothesis of Molecular Vortices,” Phil. Mae. July 1851, p. 62, supposition 3. The “ atmospheres” of his atoms appear to correspond to buttons, while their “nuclei” agree to some extent with the cord. 2B2 372 Mr. O. J. Lodge on a Mechanical illustration indicated of the close relation known to exist between electric: conductivity and conductivity for heat. They appear to de- pend on two distinct things—the first on the “ friction” between the button and the cord, the second on the elasticity of the rods or strings connecting the buttons together. Never- theless it is an experimental fact that the two apparently di- stinct things are inverse functions of one another ; not impro- bably they are, in substances of similar chemical properties, simply inversely proportional to one another. § 19. Bat now what happens when light falls on an elec- trolyte? If, indeed, the whole substance were composed, as our model ser suggest, of atoms inanearly disjointed state, it can hardly be doubted that much energy would be lost as light in effecting decomposition. But in reality these free ‘“Clausian”? atoms are very few compared with the atoms in firm ata combination at any one instant; and all the. compound molecules would be highly elastic, and would pro- pagate the light-oscillations pertectly well, the force being by no means sufficient to pull the continent atoms asunder in most liquids. Hence electrolytes, though conductors, may be transparent. It may possibly | happen that some of ie waves are to some extent destroy ed by the occasional free atoms, and that the absorption of light in deep tra nS panne liquids 1 iS partly due to this. Again, it may happen i in some electrolytes that the atoms in the compound molecules themselves are so. weakly associated as to be shaken asunder even by light-oscil- lations, especially if their periods are synchronous. ‘This dis- sociation would be especially likely to manifest itself in case . one or both of the constituents tended, as soon as free, to come out of the fluid in the solid state; and accordingly it is observed in a solution of sulphuretted hydr ogen and in Pro- fessor Tyndall’s “ actinic clouds.” § 20. If through a dielectric medium such as we have ima- gined above (§ 18) we cause an electromotive force to act in the direction of x, we must picture it to ourselves as a pulling of half the cords which lie parallel to w in the positive direc- tion, and the other half in the negative direction (see § 8, foot-note) ; ; and the elastics being stretched by this action, an x oscillation will take place about the new position of equili- briam in a shorter period than before. Let fall in the direc- ee of za beam of plane-polarized light. If it were polar- ized so that its vibrations occurred in the plane xz, it would now travel quicker through the medium than when the electro- motive force was not acting ; on the other hand, if its vibra- tions occurred in the plane yz, it would travel appar ently at 1ts ordinary rate. (In reality its velocity would in this case of Hlectric Induction and Conduction. » 373 ‘be just as much diminished as the velocity in the first case is in- creased, because of the relaxation of tension normal to the lines of force ; but this the model does not show.) Let the real plane of polarization be wy. Then, resolving each oscilla- oe into two equal components pa -allel to 2 and 2 y respectively, the y component will be retarded behind the other in their pas- sage through the medium ; and when the beam emerges, it will _-be no longer plane, but elliptically weet ‘The amount of ellipticity depends on the difference of the velocities of the two components (which depends on the square of the electromotive force), and on the thickness of strained medium through which the light has passed. If one component were retarded a quarter wave-length behind the other, the emergent light would be ‘circularly polarized; so the actual retardation must be ex- tremely small, as the change in the light is, even under favour- able circumstances, barely perceptible. This is the pheno- menon long looked for b by Faraday, but first observed by Dr. Kerr*, of Gla asgow, last year. oe College, London. Note added October 14, 1876. Since the above was in print I have seen a paper of Sir William Thomson’s in the Philosophical Magazine for June 1853, ‘*On Transient Electric Currents,” from which I gather that Weber had already applied his electrodynamometer to the investigation of transient currents—in which, moreover, I find that Thomson pre- dicted the experiment of Dr. F eddersen, mentioned above (§ 3 and footnote, p. 355), almost exactly, and calculated the value of every oscillation in a discharge. It is also shown that all the oscillations ought to occur at equal intervals of time, whatever be the resistance p of the discharging circuit, provided only that it remains constant. The interval is not indeed the simple harmonic Ar an T we : semiperiod Cie) Gey but it is Nes (nese) A being a constant which I do not yet understand, but which Thomson calls ‘“ the elec- trodynamic capacity of the discharger.” I imagine, therefore that the increasing intervals actually observed were due to the resistance of the air, across which the discharge took place, increasing as the strength and heat of the successive sparks diminished. It appears that the model agrees perfectly with Thomson’s theory, except that it locates the principal cause of the action in the * Phil. Mag. November and December 1875. The most singular point in Dr. Kerr's discovery is his observation that bodies may be divided into two classes—those which act as if compressed along the lines of force, and those which act as if extended. Ihave rep eated his experiments with glass, but not without failures sufficient to excite my admiration for the skill and patience involyed in the discovery. 374 Prof. Challis’s Theoretical Hxplanations of charged body instead of in the discharging wire; for, taking the model of a condenser of capacity C (= (he '), and discharging it through a circuit of resistance p, the restoring force, at any instant during the discharge when the cord is displaced f and is flying back with velocity u, 1S CT f— pu, there being of course no time for any slip between cord and but- tons. Writing this thus, with A for the total mass of all the dis- placed particles, acd at of=, dt and noticing that f=q, and that when t=0 q=Q and u=0, we obtain at once Thomson’s fundamental equation, (5) p. 395, from which all his results follow. In this paper I have abstained from mentioning current-induc- tion, because I have not yet read Maxwell’s second volume; but the model suggests ideas as to the nature of the process concerned in producing the extra-current &c., which I expect will turn out useful. XLV. Theoretical Explanations of Additional Phenomena of the Radiometer. By Professor CHauuis, I2A., /R.S., Iilh Reet Sn T the end of the “ Theory of the Radiometer’’ which I proposed in the Philosophical Magazine for May 1876, I stated that I gave with reservation the explanation of the rotation of the glass globe when it is floating in water, not having then seen Mr. Crookes’s communication “ On the Move- ment of the Glass Case of a Radiometer” contained in the ‘Proceedings of the Royal Society,’ No. 168, p. 409. On reading that communication, I found experimental proof that the movement of the glass globe was indicative, as I had inferred from theoretical considerations, of “ friction be- tween the glass support and the point on which the system of vanes turns.” ‘This explanation has recently been con- firmed by an experiment made by M. Jeannel (see Phil. Mag. for October 1876, p. 820), who observed that the rate of rota- tion was influenced by musical vibrations excited in the sur- rounding air, and reasonably attributes this effect to momen- tary suspensions of the friction between the pivot and its sup- port caused by oscillations impressed on the instrument by the aerial vibrations. The main purpose of the present communication is to take into theoretical consideration some additional experiments of * Communicated by the Author. Additional Phenomena of the fadiometer. aye) a very remarkable character, announced by Mr. Crookes in the ‘ Proceedings of the Royal Society, No. 172, p. 136. I refer chiefly to “the phenomena exhibited by “a al piece of pith hanging down like a pendulum at the distance of about a me from the rotating vanes of the radiometer.” It was oticed that “scarcely any movement of the pendulum was oF laced when the rotation was very rapid ; but at one parti- cular velocity the pendulum set up a considerable movement.” At the suggestion of Professor Stokes, the candle which by its light produced the rotation, was placed at the distance from the radiometer for which a revolution of an arm of the fly synchronized with a vibration of the pendulum. “In this way the pendulum was kept for some time swinging through a large are.’ To account theoretically for these facts, I have, first, to direct attention to that part of the article in the May Number where it is said (p. 396) that “the incident light thus produces an abnormal state of the atoms at and near the superticies of the vane, analogous in some degree to the state of the superficial atoms of a body electrified by friction.” Since this was written [ have seen reasons (which will presently be adduced ) for concluding that the light, or heat in the radiant form, in- cident on the vanes has the effect, after being transmuted ‘into heat of temperature, of so changing the relative positions of the atoms in a superficial stratum as actually to induce the electric state. According to the hydrodynamical theory of electricity which I have proposed in the Philosophical Maga- zine for October 1860, and in ‘The Principles of Physics,’ pp: 544-546, the electrified state of a solid body is solely and necessarily the result of a displacement of the atoms constitu- ting a thin superficial stratum from their normal positions. According to the same theory such disturbance of the super- ficial atoms is always accompanied by an interior gradation of atomic density, in consequence of which etherial streams are generated and maintained by the action of those etherial vi- brations to which are due, under normal circumstances, the attractive and repulsive forces treated of in my communica- tion in the September Number. “It was, in fact, argued in the ‘“* Theory of the Radiometer,’ ” given in ihe May Number, that the state of the vanes is such as is here stated, although they were not directly said to be electrified. I shall now assume that they are in the condition of electrified bodies, and proceed to inquire what consequences follow from this supposition rela- tively to the new facts it was proposed to account for. Since the light, or heat in a radiant form, incident on the vanes is converted into heat of temperature in greater degree 376 Prof. Challis s Theoretical Explanations of at its blackened surface than at the other, the two surfaces are electrified in different degrees ; and, relativ elv to a neutral state, one is positively electritied and the other negatively electri- fied. Now by anexperimental law of electrical action (which is also accounted for by the hydrodynamical theory), a face of the vane, whether positiv ely or negatively electrified, as it approaches the piece of pith will attract it; and the oppo- sitely electrified face of the same vane, after passing the pith, will also attract it, supposing there is no contact between the vane and the pith. Consequently the piece of pith is drawn in opposite directions in quick succession if the rotation of the vanes be very rapid ; and as impression of motion takes time, t might well happen that, under these circumstances, no per- ube motion takes place. If, however, the rotation is slow, the attraction of the vane in one direction may take effect before that in the opposite direction commences, in which case an oscillatory motion of the pith ball will be pr oduced. Clearly the oscillations will be most steady when by reason of synchro- nism of the time of oscillation of the pith with the time of re- yolution of the vane, the attraction of a given vane acts like gravity on a pendulum. If the oscillation of the pith ball be produced, as seems to be supposed, by the intervention of the action of the vanes on the residuum of air in the globe, it is wholly inexplicable that there should be no perceptible effect when the rotation of the vane is very rapid, and might be ex- pected to cause great disturbance of tLe air. ‘The facts that no oscillaticn is produced by a rapid rotation, and that oscilla- tion commences after diminishing the rate of rotation, are well accounted for by the present theory ; and these explanations justify at the same time the assumption of the electric state of the vanes. In the Institut of July 5, p. 213, an account is given of an experiment by M. Ducretet,-according to which, by throwing ether on the glass globe to produce depression of temperature, the rotation of the vanes is first stopped, and then, by continuing the cooling, is caused to take place in the opposite direction. This result is quite in accordance with the present theory. ‘The cooling effect of the ether, according to the law of ‘heat-exchanges, causes the vanes to radiate less heat, or even changes the heat into cold ; and as the cooling, for the same reason as the heating, is in excess at the blackened surface, the rotation might thus be stopped, and the rotating force might be made to act in ihe contrary direction. The movement of the vanes is affected by two causes—the presence of a residuum of air, and the friction at the pivot above spoken of. Before exhaustion of the globe, no motion Additional Phenomena of the Radiometer. 377 of the vanes takes place—the reason being, according to ue present theory, that no electricity can pass, air of ordinar density being a non-conducter. On producing echaaston the rotation commences and increases up to a certain point, because rarefied air is a conductor of electricity. After a high degree of exhaustion is attained, the rate of rotation diminishes, because vacuum is a non-conductor of eleciricity, and an approach to that condition has a retarding effect. As the exhaustion proceeds, the effect of the friction at the pivot becomes more prominent, which, however, ap peams from the experimental results to be, after all, avery minute quan- tity. A degree of exhaustion which ‘actually stopped the vanes was not reached. I can see no a priori reason why the stoppage should occur with the same degree of exhaustion as that which would prevent the passage of a galvanic spark under particular conditions. The vanes continued to move when the degree of exhaustion was considerably greater than that at which the spark ceased to pass under the particular conditions arranged for the comparison, On further trial I failed to verify the statement made in the May Number, that the presence ‘of a magnet in the neigh- bourhood of a radiometer alteeted the rate of iis motion. On the contrary, I proved by more careful experiments that the streams of a magnet had no perceptible efiect on the move- ment of the vanes. This resuit might have been anticipated from the circumstance that the electric streams of the theory traverse the vanes transversely, so that the addition of mag- netic streams (with which, according to hydrodynamics, the electric streams may coexist) produces no v ‘aviation of setheri al density at the vanes, and therefore no motive force tending to give them motion. ” The case would be different for a set of streams symmetrical with respect to an avis, inasmuch as addi- tional streams resolved transversely to the axis would, on one side of it, coincide in direction w ith the original streams re- solved in ‘the same direction, and on the other side be opposed to them, so that at the axis there would be a gradation of etherial density tending to impress motion. I think it right to explain here that the Theory of the Ra- diometer I have proposed i is wholly founded on those & priori principles (stated at the beginning of the arvcle in the Sep- tember Number) which form the basis of the theories of the different phy sical forces which | have for a long time been engaged in discussing and verifying, and that consequently it comes into no kind of competition Bann any empirical theory which the experimentalist may be able to certify by the means at his command. An established empirical theory might, how- 378 Heplanations of Additional Phenomena of the Radiometer. ever, be employed to test the truth of the a priori theory, because, if the latter be true, it ought to be capable of giving reasons for the experimental facts on which the empirical theory is founded. The phenomena of the radiometer have attracted my attention as being peculiarly adapted to be of service in carrying on researches as to the laws of physical force. I have, in fact, been able, by applying to them the. foregoing theoretical discussion, to settle an important physical question. Inmy Theory of Hlectricity [had occasion to speak of electricity-radiants, but had no means of determining in what order they stood relatively to heat-radiants. The ten- dency of the explanations given in this communication is to prove their identity, at least, as to kind, if not in degree, with heat-radiants, and to show that, like these, they are sub- ject to the law of exchanges. On the assumption that such is their character, I propose to conclude this communication with giving a theor y of the induction of electricity more pre- cise than that contained in arts. 12-16 of the “Theory of HWlectric Force ” in the Phil. Mag. for October 1860, or that in pp. 921-531 of ‘The Principles of Physics.’ This theory of electrical induction rests essentially on a cer- tain state of the atoms at and near the surface of a body (first . recognized by Poisson), according to which through an ex- tremely small thickness the density of the atoms increases from the surface towards the interior. The fulfilment of this con- dition is necessary for the equilibrium of the atoms so situated. It will now be supposed that the undulations of the ether cor- responding to the radiants we are concerned with can traverse substances freely without undergoing transmutation. Such undulations pertain to the electrified state of a solid body, and, in conjunction with the forces of atomic repulsion, main- tain by attractive action the interior gradation of density which is a necessary condition of that state. When a body in a neutral state is brought into the neighbourhood of an elec- trified body, those radiants of the latter that are incident upon it pass freely through it, and consequently traverse the whole of the thin superficial stratum of varying atomic density spoken of above. Now when ether in motion, whether the motion be vibratory or steady, permeates a collection of atoms varying as to the number in a given space from point to point, there is always an increment of the velocity of the ether, and there- fore a decrement of its condensation, towards the parts of greater atomic density. The variation of pressure consequent upon this variation of condensation acts upon the atoms and displaces them from their normal positions; and this, as we have already argued, suffices to induce electricity. Itis, how- On certain large Crystals of Enstatite. 379 ever, to be considered that this action is always towards the interior of the substance, and that the induction of electricity depends on the difference between the disturbances at the two positions at which the radiant cuts the surface of the body, which difference arises from the variation of the intensity OF the action according to the law of the inverse square. It is evident that, although the radiants may be effective in produ- cing motion of the atoms in the superficial stratum by reason of the variation of atomic density that subsists there, they may still be of an orderw hich, like light- and heat-radiants, produce no movement of the body as a whole. Cambridge, October 17, 1876. XLVI. On certain large Crystals of Enstatite found by W. C. Brogger and H. H. Reusch at Ajorrestad near Bamle, South Norway. Memoir by W. C. Broccer of Christiania and G. vom Ratu of Bonn*. ; [Plate IV.] HERE are few minerals which offer an equal interest, from the point of view of the progressive development of our knowledge of them, with the two magnesian silicates, enstatite, MgSiOs;, and olivine, Mg, SiO,. It is well known that our knowledge of the latter minera il, in proportion as it is increased, has recognized-its abundance and importance. At first found almost exclusively in volcanic rocks and in small crystals, olivine has since been met with in plutonic and meta- morphic rocks, the ‘‘ serpentine crystals” of Snarum haying been recognized as pseudomorphs after olivine. This mineral further acquires a really universal importance through its pre- sence in meteorites. Nearly the same may be said of enstatite. Its distribution and its importance have hitherto been less recognizable than in the case of olivine. Professor Kenngott, near ly twenty years ago, ave the name of enstatite to a mineral from Mount Zdjar in Moravia, the composition of which von Hauer determined as that of a normal silicate, MgSiO; (Akad. Wien, vol. xvi. p. 162, 1855). Professor Des Cloiseaux showed by means of optical researches that the crystalline system of enstatite is rhombic, and established the difference between augite and enstatite (Bull. Soc. Geol. vol. xxi. p. 105). Professor Rammelsberg first proved by chemical analysis that enstatite is a constituent of meteorites, the stone of Bishopsville (onatsber. Akad. * Communicated by the Crystallometric Association. Received Octo- ber 9,—Read October 27, 1876. 380 MM. W.C. Brégger and G. vom Rath on certain Berlin, 1861). Professor N. Story Maskelyne eas this mineral in the Busti meteorite (Trans. Roy. Soe. clx. 189). Professor von Lang showed, in an admirable iny estigation, how rich were the Eombaons of the enstatite of the Breiten- bach meteorite. Nearly at the same time vom Rath deter- mined the crystals of hypersthene of Laach. The specimens from both these sources, cosmical and terrestrial, have abso- lutely the same angles. "Implanted crystals of enstatite pre- senting freely anal crystalline faces had hitherto been un- known in plutonic rocks. It might be thought that if they could be found they might cnn us by hee gigantic size, like the olivines of Snarum; and in this expectation we are not disappointed. In the apatite mine of Kjorrestad between Kragerde and Langesund, in the autumn of the year 1874, enstatite was discovered in crystals of a size reached only by very few mi- nerals. The locality where the great enstatite crys stals have been found is one of those numerous apatite veins of Southern Norway. The main rock of this part of the coast is mica and amphibole slate, in which the apatite veins are included. Their predominating mineral is amphibole. In the eee of these normal veins there is to be found at the Hankedalsvand, not far from Vestre Kjorrestad, an isolated Tepes as a thick lode constituted principally of large crys- tals of enstatite and enormous masses of rutile. There was not much apatite, but some green ish-white mica and tale ; and for this reason apatite was obtained only for a short time._ The enstatite crystals, from 0°3 to 0-4 metre in size, had been thrown away unheeded, . they were discovered by Brégger and Reusch in hen esearches on the apatite beds (Zeitschr. d. deutsch. geol. Cpsellach vol. xxv. p. 646, 1875). As the mine of Vestre Kjorrestad had been already abandoned, the investigations of the discoverers were confined to the matter thrown out. The enstatite crystals, occurring in more or less elongated columnar forms always broken at one end, were no doubt origi- nally implanted on the wall of the lode. The space between the giga ntic enstatite crystals was filled with silver-white or hight-green talc. Also in the interior, and principally in the decomposed crust of the enstatite, we see minute scales of igs lying in planes parallel to the prismatic faces, or oftener to the brachypinakoid, formed evidently by a metamorphosis of the enstatite. The new crystals excite our peewee at first by their size. Several crystals attain a magnitude of 20 centims. in len eth co) and in breadth. One of the two largest specimens measures large Crystals of Enstatite. 381 38 centims. in length, 26 in breadth, 13 in thickness ; the other one, notwithstanding its being broken at both ends, is still 40 centims. long. The crystals show a predominating vertical rhombic prism, whose edges are nearly rectangular. The makropinakoid is rauch developed, whilst the brachypinakoid is only small. An evident rhombic symmetry is seldom to be recognized on the summits of the crystals ; it offers, on the contrary, a pseudo- monoclinic appearance. ( ae G(a, y, 2) and B=G(A, k, 1). G(M, ke’, V). It is further noticeable that if F and G are contravariantive forms, each numerator of the fractions expressing the differential derivatives of t is nullified by the operator V G(x, y, 2) F d.-d “ad. (i dy dz} and conversely, every rational integer function of , y, z so nulli- fiable is a linear function of such numerators. And so in general the Theory of Spherical and Prospherical merges in a theory of Conicoidal and Proconicoidal Harmonics.—J. J. 8. Steamship ‘ Parthia,’ Sept. 8, 1876. THE LONDON, EDINBURGH, axn DUBLIN PHILOSOPHICAL MAGAZINE AND JOURNAL OF SCIENCE. [FIFTH SERIBS.] DECEMBER 1876. L. On Bi- and Unilateral Galvanometer Deflection. By G. CurystaL, B.A., Fellow of Corpus Christi College, Cam- bridge™. af WAS led to study the subject of this paper during a series of experiments undertaken for the purpose of directly testing Ohm’s law. | The results arrived at are, I think, interesting, not only in connexion with galvanometry, but also in relation to the theory of induced magnetism. In the first instance I shall describe the phenomena as simply as possible from the first _ point of view, and then consider a little more closely some points which arise when the maiter is looked at from the second point of view. This order is to a great extent that in which the facts came under my notice ; and it has the additional ad- vantage that it leads us incidentally to see that the phenomena in question really have their seat solely and entirely within the galvanometer, and have nothing to do with any phenomenon of the nature of unilateral conductivity or with any other ex- ception to Ohm’s law. Dr. Schuster has describedt an experiment in which a _ small current of constant direction is superposed on the alter- nating currents of a sine inductor, and the whole sent through a galvanometer. Such an experiment affords (under certain suppositions) a test of Ohm’s law ; for the average intensity of the current in the direction of the small constant current is greater than that in the opposite direction ; hence, if the re- * Communicated by the Author. + Phil. Mag. [IV.] vol. xlviii. p. 340. Phil. Mag. 8. 5. Vol. 2. No. 13. Dec. 1876. 2D 402 Mr. G. Chrystal on Bi- and Unilateral sistance of the circuit depends on the intensity of the current, one of the currents will prevail, and the presence of the alter- nating currents will affect the permanent deflection due to the small constant current, whereas, if Ohm’s law were true, there would be no such effect. Without discussing Dr. Schuster’s results here, we may remark that a similar test of Ohm’s law could be obtained by merely passing through a galvanometer the currents from the secondary coil of an in- ductorium. It is well known thatif the primary be made and broken periodically, there will be an alternating current in the secondary, which will have the same period. The whole quantity of electricity which passes during a complete oscilla- tion is zero ; but the maximum intensities of the positive and negative parts of the current are very different. The positive part, starting immediately after the break, has a considerabie initial intensity, which is independent of the resistance of the secondary; the negative part, starting at the make, begins, on the contrary, with zero intensity, and never reaches so large a maximum as the positive™. It follows that if the resistance of the circuit depends on the intensity of the current, then the two parts of the current will not experience equal resistances, and we shall get a galvano- meter indication in the direction of that which has the advan- tage. Any such effect would be much increased by the in- troduction into the secondary circuit of a resistance composed of very fine wire. It is easy encugh to calculate what this resistance should be, in order to produce the greatest effect on - the galvanometer. Such a resistance I used in the shape of a fine German- silver wire (‘002 inch diameter) wound on a cylindrical piece of vulcanite about 9 inches long, the turns being insulated from each other by the thread of a screw of one hundred turns to the inch cut in the vuleanite. The whole is enclosed in a glass tube with brass caps and copper terminals. or this in- strument I am indebted to Mr. Garnett, of St. John’s, De- monstrator at the Cavendish Laboratory. The induction-coil used was of the ordinary lecture-room form by Apps; the primary was made and broken by electric tuning-forks of various pitch. | The results I obtained indicated an apparent departure from Ohm’s law, sometimes in one direction, sometimes in the other. The presence or absence of the fine wire in the circuit did not seem to be an essential condition of the phenomenon. * This supposes the period of alternation long compared with the time- constants of the coil; the same description applies, in a modified degree, to other cases. Galvanometer Deflection. 403 T was therefore led to suspect that the cause lay in the galva- nometer itself{—a suspicion which became certainty when I found that reversing the galvanometer-connexions with the secondary, or reversing the primary, had no effect whatever on the character of the phenomena. It appeared that the effects observed could be analyzed as follows :—Suppose we are using a Thomson’s galvanometer with mirror, lamp, and scale as usual, and let the scale be placed parallel to the coil- windings, a common perpendicular passing through the centre of the mirror, and the slit through which the light comes from the lamp ; then two distinct states appear, according to the relation between the strength of the alternating currents in the secondary and the strength of the magnetic field in the axis of the coil due to the earth and deflecting magnet. Ist. If the currents are powerful enough and the magnetic field weak enough, the spot of light goes ; off the scale completely, either to one side or the other, and remains there. It can be made to go to either side and remain there by starting it off properly, which is easily enough managed by throwing on the alternating currents after it has passed the zero towards the side to which it is desired to send it. The spot will not re- main at zero, even when placed there very carefully. This phenomenon | call bilateral dejlection*. 2nd. If the strength of the currents be decreased sufficiently, whether by interpola- - ting resistance in the secondary or primary, or by reducing the electromotive force in the primary, or by shunting the gal- vanometer—or if, on the other hand, the strength of the magnetic field be sufficiently increased, say, by lowering the deflecting magnet,— I. If the spot of light be brought, when there are no elec- trical oscillations, to zero on the scale, then on setting the coil in action it comes to rest at zero and remains steady there. II. If the spot be brought to any position right or left of zero, then when the coil is in action it comes to rest in some position a little further to the right or left respectively, and remains steady there. The difference between these positions is greater the greater the origina! deflection of the spot from zero. ‘This phenomenon I call unilateral deflection. IIL No difference of any kind was produced i in any of these phenomena by reversing the connexions of the secon- dary with the galvanometer. Nor did the character of the phenomenon depend on the number of alternations per second, which in my experiments varied from 10 to 200. It was the second of the last-mentioned set of phenomena * After Pogeendorff, who originally observed the phenomenon and called it “ Doppelsinnige Ablenkung.” pee ae vol. xlv. p. 3853 (1838). 2 S 404 Mr. G. Chrystal on Bi- and Unilateral that first attracted my attention ; and the reason which seemed to me to explain this suggested the existence of the first set (I was not then aware of Poggendorft’s description). Fact III. shows that the phenomena have nothing to do with any departure from Ohm’s law in the case of induction-currents ; and it seemed clear that the cause must be sought for in alterations of the magnetism of the needle. Bilateral deflection was observed and minutely experi- mented on by Poggendorff. Unilateral deflection does not seem to have come under his notice, and has not, so far as I am aware, been described elsewhere*. Poggendorff gave a general explanation of the phenomenon he observed, attribu- ting it to the effect of the alternating currents on the mag- netism of the needle. His description has led others to sup- pose that the effect never appears otherwise than in rendering the needle unstable f. It will be seen, however, from the above facts that this is not so. It is easy, mcreoyer, to show that a general theory of the phenomenon would predict this. Let @ be the inclination of the needle to the plane of the galvanometer-coil windings when no currents pass, @ its incli- nation at any time; then the magnetic couple tending to bring the needle back to its position of rest may be represented by sin (@—a), supposing there were an alteration of the mag- netism in any position proportional to the magnetic force due to the current resolved parallel to the needle (a more accurate statement will be given below); then the couple due to this tending to drive the needle from its position of rest would be > proportional to B sin 26, B depending on relative strengths of electrical oscillations and magnetic field ; thus the force tend- ing to bring the magnet back to position of rest would be pro- portional to 7 sin (@—a)—B sin 26. (We neglect the inductive action due to the permanent mag- netic field.) Suppose we draw on two pieces of thin paper the curves y= G, 3) 2) ROE ey ee er and y =B sin 20.1) ae ee Then, by superposing the pieces of paper and looking through them towards a light, we can see where the curves intersect * Since writing this paper I have learned from Lord Rayleigh that similar phenomena have come under his notice. Beyond Poggendorft’s, I know of no published account of the matter. + Schuster, Phil. Mag. [IV] vol. xlviii. p. 257. Also Wiedemann, Gal- vanismus, Bd. ii. 2, p. 284. Galvanometer Deflection. 405 for different relative positions. These points of intersection correspond to the positions of equilibrium of the needle ; and the stability or instability of the equilibrium is seen at a glance. For example, take the case «=0. The points corresponding to @=0 in (1) and (2) must be superposed ; thus, for a large value of B we have fig. 1. Here O, P, P’, Q, and Q’ give Fig. 1. positions of equilibrium ; but O is obviously unstable, because on going to the right (increasing @) the curve (2) lies above the curve (1), 2. e. the force tending to increase @ preponde- rates ; and similarly, if we go to left (decrease @), the force tending to decrease 0 preponderates. Similarly, Q and Q’ are unstable positions ; but P and P’ are stable positions symme- trically situated with respect to O. The positions of P and P’ lie nearer a point distant + from O, the greater B. This case is best illustrated experimentally with a tangent-galvano- meter of the usual construction, where the motions of the needle can be traced all the way round. We have thus ex- plained Poggendorft’s case, which is the limit to the state of the phenomenon”. Next, suppose B to be small, we have fig. 2. The positions Fig, 2. “ There are some points in Poggendorff’s observations which did not appear in mine ; but the difference might arise from the inductive influence of the earth being sensible in his experiments, which was not the case in mine, 406 Mr. G. Chrystal on Bi- and Unilateral P and P’ have now disappeared, and the equilibrium at O has become stable. This corresponds to case (1) of the second state of the phenomenon. Lastly, suppose B small and # not zero, then we must dis- place the origin in curve (1) to the right, say. We thus get fig. 3. Here O O’=a, so that O’ represents the position of Fie. 3. equilibrium when there are no currents. P and P’ are un- stable ; and Q represents a single stable position when the cur- rents are going. ‘The deflection corresponding to this posi- tion is On, where n is the foot of the ordinate of Q, and O nis > O O’, in accordance with the experimental facts above stated. Similar reasoning would show that if the magnet had been deflected in the opposite direction, the alternating currents would have increased the deflection in the same direction. In fact the above simple graphical representation embraces the experimental facts, as I have observed them, completely. It may render the above clearer to give the results of one of the earlier experiments with a Thomson’s galvanometer. No care was taken to adjust the scale parallel to the windings of the coil; so that the results are not symmetrical ; but they illustrate perfectly the nature of the phenomenon in its second state. Position of spot Position of spot with no current. /with currents going. bese: +173 +350 177 + 50 +108 58 — 2 + 10 12 — 14 — 15 1 — 20 oO ta 3 10 — 64 —134 70 —193 —355 162 In this case the mirror was very nearly parallel to the coils when the spot of light was at —14 on the scale. It will be seen from the above that the effects are of a very decided and unmistakable character. I may also add that I have got similar results with three different galvanometers. Galvanometer Deflection. 407 This may be sufficient so far as the subject is connected with galvanometry merely. I need hardly call attention to the im- portance of these phenomena in relation to experiments such as those of Dr. Schuster, where alternating currents are used. In attempting to get a clearer insight into the nature of the effect on the magnetism of the needle, I was led to make some numerical verifications, and to give some variations to the experiment which may perhaps be of interest. The phenomenon of magnetic induction is more or less complicated according to circumstances. We have to deal, in fact, with several distinct phenomena, which may for the present purpose be classified under two heads :—I. Tempo- rary Magnetism; II. Residual Magnetism and its gradual decay. These phenomena are analogous to (1) the temporary strain, (2) the permanent set and “elastic recovery ’’* in a solid subjected to stress in any way. In what follows I shall assume that we have to do with temporary magnetism merely. Hven when thus simplified the present case is to some extent peculiar. It is probable that the maximum magnetization producible by a given force is attained only after the force has been in action for some time f. Now, ‘if the effects we are considering be due to in- duced magnetism at all, it is obvious that a very considerable fraction of the induced magnetism due to a given magnetic force musi be developed in an interval of time incomparably smaller than the <4, of a second, whereas in the ordinary experiments on induced magnetism the time allowed for the development of the magnetization is practicaily unlimited. This peculiarity gives the present case additional interest. I shall, in what follows, assume that when the magnetic force to which an element of iron or steel is subject varies, the corresponding variation of the magnetic moment of the element follows at an interval of time which is incomparably shorter than any other we have at present to deal with (e. g. the time during which either of the induced currents remains in the neighbourhood of its maximum). With this assumption, we may apply the ordinary theory of magnetic induction. Three general conclusions may at once be drawn from the simple consideration that an elongated body tends to place its axis parallel to the lines of magnetic force :— * By elastic recovery is meant what in Germany is familiarly known under the name of “ Elastische Nachwirkung.” I do not know of any English name for it which has the scmenten of ‘good authority. * Wiedem: ann, Galvanismus, Bd. II. 2, p. 160. Also Faraday, Phil. Mag. [IV.] vol. ix. p. 92. 408 Mr. G. Chrystal on Bi- and Unilateral I. An elongated magnet magnetized axially would give phe- nomena analogous to those observed. il. A spherical magnet would give no such phenomena. III. An elongated magnet magnetized transversely would give similar phenomena, ‘except that in unilateral deflection the sign of the effect would be reversed ; 7. e., the spot of light being “brou ght, by means of the deflecting magnet, to the right or left of ZeYO, The effect of the alternating Bia: would be to diminish Ta deflection. Conclusion III. was directly verified ; and the result was in complete agreement with theory. Conclusion IJ. was also verified experimentally. A small spherical steel magnet was fitted with a mirror hung up in the galvanometer and observed, as will be afterwards described in the case of an elongated magnet magnetized axially. The magnetic moment of the sphere was roughly determined for me by Mr. Shaw, B.A., of Hmanuel College ; the maximum horizontal earth-couple on the sphere was about ‘22 grm. = = Two cells of Grove were ea in the primary ae the induc- tion-coil. The result was a feeble wnilateral deflection. An observation was made with an elongated magnet con- sisting of a piece of thin watch-spring magnetized longitudi- nally. The maximum earth’s couple in this case was. about cm. ‘27 grm. (ey All the other arrangements were exactly a before. The result was strong bilateral deflection. Here the two magnets were very nearly in the same circumstances, the ad- vantage being somewhat in favour of the sphere, owing to its smaller moment. It appears, then, that the form of the mag- net has a very powerful effect on the phenomenon. That the effect should be absolutely nil with the sphere was not to be expected; for we know™ that a piece of steel once permanently magnetized by a force F is never in the same state as it was originally. We can demagnetize the steel ap- parently completely by a force less than F'; but it requires a force greater than F to magnetize to an equal degree in the opposite direction. The particles of a magnetized steel sphere have therefore a quasi-crystalline structure related to the mag-— netic axis; so that the perfect symmetry which causes a sphere to behave neutrally in a field of uniform force, as far as induced magnetism is concerned, is probably in some degree lost, and can only be restored by ‘heating the steel over red heat and * E.g. Rowland, Phil. Mag. [IV.] vol. 1. p. 358. Also Fromme, Pogeg. Ann, Eerghd. vii. p. "421 ; Phil Mag. [IV. ] aa 1, p. 299. . Galvanometer Deflection. 409 allowing it to cool apart from magnetic influence, or by some other equivalent process of molecular revolution. | A closer examination into the behaviour of an elongated magnet further confirms the above theory. Let us take as type of such a magnet a very elongated ellipsoid of revolution magnetized parallel to its axis, and suspended from a point in its equator. Let o£ be the plane of the galvanometer-wind- Fig. 4. ings, oF the direction of the resultant magnetic force (I) when there is no current, o X the direction of the axis of the magnet at any time, a= Hob, koe. m= permanent magnetic moment, « = coefficient of induced magnetization, V= volume of magnet, 7= current at any time ¢, g = constant of galvanometer, nm = number of breaks per second in primary. The component forces parallel to oX and oY at time ¢ tending to magnetize the needle inductively are X=gisin + F cos (@—«), Y=gicos 0—F sin (9—2a). Hence (see Maxwell’s ‘ Electricity,’ vol. ii. pp. 65 & 67) the couple tending to increase @ is ai V («— ts) XY =" — (97? sin 20 + 2F yicos(20—«) 1+ 27K 1+27K _ F sin 2(9—a)}. 410 Mr. G. Chrystal on Bi- and Unilateral Considering now what happens during a complete oscilla- tion, let P be the uniform force whose action during that time is equiv: alent to the action of the varying force due to induced magnetism, then E = {i in2of ead = sin 200-2) } The middle term disappears because (" idt=0. Hence, if P? 1 eo 0 denote ( edt ==, i, €. the mean square of the induced cur- “0 rents, then P=A/T’ sin 20—B’/sinQ(@—a), . . (1) where TeV 9 igs aK V EF 1+ 27k’ ~ 142K Adin now the couple due to the permanent moment of the needle, we get for the whole force tending to decrease @, mF sin (@—a)—A’T sin 204+ B’sin2(@—a2), . . (2) In all the experiments of verification to be afterwards de- scribed (and in all the experiments discussed in this paper) the permanent field on the galvanometer was very much weak- ened by properly adjusting permanent magnets. Under these circumstances B/ is very small compared with mF and A/T. It is easy to verify this by comparing the observed values of | the times of oscillation of the needle when the currents are going and when they are not, with the values calculated by means of (2). The verification, however, is suppressed, as it is long and uninteresting. The expression for the couple therefore becomes mi sin (@—a)—A’T?sin 20, . . -. ap the same as that obtained above by looser reasoning. It ap- pears, then, in the first place, that if we make a series of observations of unilateral deflection, and if a define the posi- tion of the needle for no current, and @ the position when the currents are going, then, other things being equal,. sin (0-72) 2 A’? sin 20, | ant To verify this law I used the following arrangement :—The galvanometer (designed by Professor Maxwell and originally made by Warden, Clark, and Muirhead, but rewound by myself last summer) has two coils wound in channels = inch broad, 1, inch deep, and 4 inches in external diameter. The number A/= = const =C say. Galvanometer Deflection. 411 of windings, partly of thin and partly of thick wire, is about 2668, making up a resistance of about 68 ohms. The chan- nels are cut in the same piece of boxwood, along the axis of which is drilied acylindrical hole in which hangs “the magnet. The ends of this cylindrical cavity are closed by two caps, one of which is fitted with a plano-convex lens, the other with a piece of plane-parallel glass. The torsion-head is fitted with a Weber’s suspension-screw in the usual way. The whole or the upper part is supported on a foot, with placing screws in such a way that the coils can be turned about a vertical axis. The magnet consisted of a piece of silvered glass fitted to a brass frame, with weights for increasing the moment of inertia. To the back of this was fitted a thin piece of magnetized watch- spring about 10 millims. long and 2 millims. broad. Under the earth’s force the period of the needle’s oscillation was 22:2 sec. This was raised to 49°2 sec. by properly weakening the field. The deflection of the magnet was measured by means of a scale and telescope in the ordinary way, the position of rest being deduced from three elongations. The deflection of the coils was measured by means of the faint i image of the scale from the plate-glass cap which closes the cylindrical core. The battery used was six cells of Smee; and the primary circuit was made and broken by means of an electric tuning- fork lent me by Mr. Dew Smith, driven by the primary cur- rent itself. The induction-coil was of the ordinary construc- tion, the resistance of the secondary being about 2714 ohms. By making an observation with the planes of the plate-glass cap and mirror very nearly parallel, the value of the correction for the deviation of magnetic axis of needle in this position from the true plane of ‘the coils was found. Using this cor- rection, a series of observations with different deflections of the coils gave the following result :— a, 0. C, O° i ° i +1 28 1 55 1175 +4 0 3 13 1172 +5 0 6 29 "1156 +5 16 6 54 “1195 —4 34 6 3 "1225 Considering the means employed in obtaining them, these values of C, differing from the mean by less than 4 per cent., agree as W ell with each other as was to be expected. The tw 6 disturbing elements were the inconstancy of the battery and 412 Mr. G. Chrystal on Bi- and Unilateral the varying residual magnetism of the iron core in the induc- tion-coil. I next made some experiments to determine whether, other things being equal, C varies as I’, which by the above theory it ought to do. With this object in view different resistances were interpolated in the secondary, every thing else being kept the same. The resistance of the secondary, including the galvanometer, was about 2768 ohms. Resistances of 1000 and upwards were put in, « and @ observed, and the resulting values of C calculated. If we suppose the time which elapses between two successive interruptions .of the primary to be so long that the current in the primary arrives at the steady state in the interval, and the induced currents in the secondary due to make and break do not interfere, then it is very easy to calculate the value of I’ for the induction-currents. The result is n MP7? NP 2NQ i. * LOsNP IO. (cee where L, N are the coefficients of self-induction for the pri- mary and secondary, M the coefficient of mutual induction, and P and Q the respective resistances, 7 the steady current in primary, and 2 the number of interruptions per second as before. Of the two terms within the bracket the first is con- tributed by the current due to the break, the second by that due to make. Now with an induction-coil such as I used—where L=:013, M=-‘79, N=52, and P=2 (say) and Q =2768 (these num- bers are very rough estimates deduced from experiments per- formed for practice )—the time-constants of the coil are such that with tuning-forks such as I used for producing the break (which gave n=50, 100, or 200) the above formula is very far from being applicable. In fact the result is much nearer what we should get by assuming that the primary current followed the sine law, in which case we should get, A being the maximum electromotive force in primary, and v=27n, Pe 4YM?A? B = (N= IP) + OPPs EP QS Lye Pq” * in which case it is easy to see that if v be big enough, the effect on I’ of doubling and trebling the resistance Q will be comparatively small. This is confirmed by experiment, as the following Table will show :— — Galvanometer Deflection. 413 — Values of C for different values of 7. Q. 50. 100. 200. 2780 "2448 2211 "2128 3780 2359 2126 2062 8780 1798 "1829 "1822 12780 1424 "1568 1657 When the value of 7 is less, the value of C ought to fall quicker as Q increases. This is confirmed by the following result of a series of experiments in which n had the value 10 very nearly :— Q. C. D. 2780 ‘0774 1237 5480 0475 1637 10480 0242 1774 12780 0194 1796 The column D gives the first four figures of the reciprocal of if { 1 NP CQL T NP+LQS’ which ought to be constant if I? followed formula (A). It will be seen that D is not very far from being constant, the differences getting less as Q increases. This is what we should expect. : If instead of using the induction-coil in the usual way we throw the battery into the coil usually used as the secondary, and put the galvanometer into the small-resistance coil com- monly used as the primary, then the induced currents, even with the 200 fork, will not interfere. We must in this case put in formula (A), which is now perfectly applicable, 7 M79 N=-013, | P=2720\ (say), Q=68, the last arising practically from the galvanometer resistance. Hence NP LQ+NP and when Q is doubled its value is ‘005 ; so that the fraction contributed by the current at make to the value of I? is now comparatively small. Hence C will approximately vary in- versely as the resistance Q, and C ought to drop to nearly half its value when Q is doubled. The experiment was tried with twenty-five small Leclanché cells in the secondary of the induction-coil ; the breaking-fork =-(10, 414 Prof. E. Ketteler on the Dispersion of Light was now driven by an auxiliary battery. The value of n was 50. The galvanometer (resistance about 68) was put in the primary as above described. The above calculation represents the case thus realized pretty closely ; for although the self-induction of the galvanometer has been neglected and the resistance of the battery only roughly estimated, yet neither of these affects the important term. The result, therefore, of experiment ought to be nearly as above predicted. It was so as nearly as could be seen, taking account of unavoidable experimental errors. It appears, therefore, that the above theory stands so far the test of experiment. When I can get the use of a sine-inductor or a sufficiently delicate electrodynamometer (both of which will probably soon be added to the collection of instruments at the Cavendish Laboratory), it will be easy to test the theory still further. If it be accepted, it seems to me that an interesting conclu- sion follows, viz. that, of the total induced magnetism which a given field of force is capable of generating in any body placed in it, a very considerable fraction must be developed in a time very much less than z}>5 of a second. Perhaps a method for measuring the inductive capacities for temporary magnetism of strongly magnetic substances might be built-on the experiments I have described ; but this can hardly be done until it is better known what degree of accuracy can be ascribed to the law Cicale: Possibly by sufficiently increasing the speed of revolution we might with a sine-inductor be able to introduce the element of time into magnetic measurements, and thereby get new light on the difficult subject of magnetic induction. Cavendish Laboratory, Cambridge, October 2, 1876. LI. Attempt at a Theory of the (Anomalous) Dispersion of Laight in Singly and Doubly Refracting Media. By Professor i. Kerrecer. {Continued from p. 345. | 7. % EF we now make the attempt to extend our theory to an- isotropic media also, only one procedure will lead to the end in view, and that totally different from the usual one. Ac- cording to Fresnel’s method, namely, the mathematical treat- ment has hitherto been restricted exclusively to the differential equations of the vibratory motion of the xether perpendicular to the normal of the waves; and by means of them the in Singly and Doubly Refracting Media. Ald velocity-surface of the normal (corresponding to the “ first ellipsoid”’), as the primary, has been derived. Making use of this latter, the velocity-surface of the rays, the ‘“ wave- surface ”’ (corresponding to the second or Pliicker’s ellipsoid), is then obtained as the envelope of the same. The wave- surface thus appears only as secondary or derived from the former, while in nature it is quite the reverse: here the wave- surface only has a physical meaning, and the normal surface is associated with it solely as an (of course valuable) auxiliary surface. The incorrectness referred to can only be avoided by ad- mitting also the vibrations of the corporeal particles. Let the medium whose doubly-refracting properties are in question have resulted from an isotropic one, with molecular distance 7, equai on all sides, through being exposed in three perpendicular. directions to the pressure- or pull-forces Px) Py, Pz, and thereby attained lmear extensions exactly pro- portional (as we will assume) to these pressures, which then have for their consequence the coordinate-distances c=x,(1+ APr)=x(l+a), y=y(l+apy)=y(1+8), 2=2(1+apz)= z(1+y). The variable molecular distance for any direction whatever, which forms with the axes of pressure the angles a, b, c, is calculated therefrom (as I will further on show) by means of the equation cos" cos"b cose ro (1 +a)? : irae) i Fy (ly) If now an ether-point of the medium is permanently shaken by any external force, this motion is propagated to all the - surrounding zthereal and corporeal particles; and after the lapse of, say, the unit of time, it has proceeded as far as a surface called cat’ ¢Eoynv the wave-surface. Along each radius vector of this surface (a “ray’’) the ethereal and corporeal particles are therefore in associated motion. The condi- tions, however, of this association are, in my opinion, the following :— a. The vibrations of the ethereal and corporeal particles necessarily take place in the plane given by the ray and the wave-normal, which therefore at the same time appears as a certain plane of symmetry of the medium. b. The vibrations of the ether particles (which latter, on account of their minuteness, we conceive as at least approxi- mately continuous) lie, by virtue of the incompressibility of . the particles, within the tangential wave-plane. c. On the contrary, the force which results from the resist- ance of the more discretely situated corporeal particles stands iL (a 416 Prof. E. Ketteler on the Dispersion of Light perpendicular to the ray; and so far they do not behave otherwise in anisotropic than in isotropic media. d. On the other hand, again, on account of the difference intimated, it does not appear indispensable that the corporeal and zxthereal particles should vibrate in parallel directions. We preliminarily leave it undecided whether the vibrations of the former, so far as they have generally a regular direction, are perpendicular to the ray or to the normal. This supposed, let e be the constant of deformation of pure ether; and let H, H’, K be the constants of the additional forces, originating from the resistance of the corporeal parti- cles and acting perpendicular to the ray. Consequently, as regards the equation of motion of the ethereal particles vibrating perpendicular to the normal, we have, if e acts exactly in this direction and to e the increment Ei cos 6 (understanding by 6 the angle between the ray and the wave-normal) is added, m™P = (e+ Ecosd)eP (15 4) Cie dit a ee On the other hand, for the corporeal particles the previous differential equation 2 Af B) If “2 mo Woe Kp aes remains afterwards as before; and in it HK’ and K are to be referred to a direction perpendicular to the ray. We also again make the assumption that Hae, Hae Kenn 2 eee To integrate these equations, we imagine the actual excur- » sion p (A) as a component of the virtual excursion p,(A,=4), and put tes beagles Po=A cos 2n( a +7 o)- 5) eer Pp =A cos an( a+ oy Here, consequently, as may be particularly remarked, the abscissee # also are referred to the direction of the ray; and just so the // are the internal wave-lengths measured in the same direction, not in the direction of the normal. If, finally, we denote the ray-velocity by o (in contradi- stinction to the wave-velocity w), and put, besides the sine- ; sine Vv ane . ratio n=———=--,, the ratio of velocities n’= _, we obtain for sinr @ o in Singly and Doubly Refracting Media. A17 the latter, analogous to what was obtained before, E ad Cos ) re eh (18) ee ss z oF mA gehen PS To if we generalize as much as possible by introducing the symbol of summation. 8. The problem that now remains is, to express the values =, == as functions of the variable molecular dis- tance 7; while m/ and m remain, as masses of cubic space, independent of any orientation. It may first be asked, Which linear density of the corporeal structure comes into consideration in relation to the resistance to the vibrations of the ether? that in the direction of the giving-way of the corporeal particles, or that in the direction of the ray, or that in a third direction perpendicular to the two mentioned? We shall unhesitatingly select the first. Further, e’ and « both depend generally, complementary to each other, on the form, the chemical quality, and the forces of the molecular combination. It hence appears probable that every alteration”of density will affect the one quantity as well as the other. In fact the result of my previous memoirs is, if that the quotient set is not merely independent of the cubic density for gases and liquids, but also has an identical value for the two or three principal indices of refraction of anisotropic media. On the other hand, it is different for calc- spar and aragonite notwithstanding their similar chemical composition. We shall therefore regard our constant L? as connected solely with the optico-chemical quality. re : The quotient —, on the contrary, as the ratio of two quan- € tities belonging to one another, of deformation of the sxther and the corporeal particles, will necessarily change with the molecular distance 7 of the latter. Now, since for 7 infinitely great (which of course implies that m/=0) 7 is equal to 1 and an increase of n/ is united with the diminution of 7, there- fore = will be inversely proportional, at least approximately, to some power of this distance. We select, for obvious reasons, the first; and if we thus put € b i cama e ° ° e . . ° (19) Phil. Mag. 8. 5. Vol. 2. No. 18. Dec. 1876. 2 418 Prof. E. Ketteler on the Dispersion of Light understanding by 6 an absolute constant, we have experiment /2 7 = in the grand on our side, inasmuch as it is known that 7 total changes but little with the density. Therefore only the value of 6 still remains to be cleared up. For this purpose we will imagine, the following experiment realized :— Upon the plane dividing-surface of an isotropic substance falls, at the angle of incidence 0, the plane of a linearly polarized wave—that is, a pencil of an infinite number of parallel rays. Tt will enter the interior without refraction; and its polariza- tion will remain, within as without, the same. Of the pene- trating rays we take one, and imagine the corporeal particles with which it comes into contact in its path characterized by some external token. We then compress or dilate the medium in two directions erpendicular to one another (but which, for simplicity’s sake, shall both be parallel to the vibration-plane) unequally. The result is twofold. On account of the unequal axial extension, all the rows of molecules which do not fall into the direction of this axis are rotated a certain measurable angle, and among them the line before indicated, whose previous angle y, with one of the force-directions changes into y¥,-+6,=x. Secondly, the previously singly refracting medium becomes optically uniaxal for the plane considered ; the refracted (extraordinary ) wave-plane now corresponding to the incidence-angle 0 cer- tainly remains parallel to the incident wave-plane; but the ray belonging to it appears, with respect to the incident (y,), likewise rotated through some angle 6. Thus, in consequence of the modification taken, with one and the same line of space x, the two new directions y,+6,=x and y,+6=¥/ would be associated. If at first one of the axes of pressure is made to coincide with the incident ray so as to make y,=0, then 6=0/ also becomes =0. If y, be then increased, 6, and 6 will simultaneously increase : for the vicinity of y,=45° they reach their maximum, and sink again to 0 for y,=90°. For this peculiar behaviour of the two directions y and x’ (one of which is, besides, conditioned by the coexistence of the other) there is, in my opinion, no other satisfactory solu- tion but just the postulate y=’, 6,=6. According to this the angle 6 between wave-normal and ray (or the virtual vibration-directions corresponding to them) would be the same as the angle between the former direction as that of some particles of the unmodified medium and the direction of the same particles after the modification. Or, in other words, The internal ray corresponding to one determinate external in Singly and Doubly Refracting Media. 419 wave appears to be connected with the cooperation of the iden- tical corporeal particles, whether the medium be isotropic, or by external forces transferred into the anisotropic state*. The inference from such external forces to molecular forces is sufficiently obvious. If, moreover, the medium is a compound, so that the sum- mation-symbol of equation (18) comes into use, we may provisionally content ourselves with the special case that 6 is the same for all its optico-chemical elementary constituents, and therefore cos 6 can be placed before the symbol of sum- mation. 9. Assuming this, r and 6 can be calculated in the following manner :—Griven the distance R, of any particle of the un- modified medium from the origin of coordinates ; let the line which joins the two (perpendicular to the above-noticed row of molecules) make with the axes the angles a,, b,,¢. We have then a y z 0 0 O20 cos b) = cose =a y] 0 ) R, Ro Ry In consequence of the modification the same row of mole- cules arrives at the somewhat different position a, b, c, and the distance R, becomes R. Correspondingly there are then COS Ay)= cos a= a cos b= Z cose= =. For 5, R, and R, the rela- R R R tions now hold good :— cos 6= cos a, cos a+ cos bh, cos. b+ COS Cy COS € 5 Reet+yt2; Raat +ye+2. Now according as, by means of the axial distances of § 7, we reduce either the angle of the new direction to that of the old, or the angle of the old to that of the new, noticing the proportionality of R, R, to 7, 75, we get the following expres- sions :— 7 cosd =r,(1 +) cos’a, +7)(1 + 8)cos’hy +7)(1 +7) c08"¢,) r?=75(1 +)? cos? a, +r9(1 +8)” cos’d, + ro(1 +7)?c087ey3J ) * To the above consideration the theory of aberration (Ketteler, Astron. Aberrationslehre, p. 177) presents the following analogy :—Let a ray AB, incident upon an isotropic or anisotropic medium at rest, produce in its interior the refracted ray BC ; and let the three points A, B, C be imagined. as fixed once for all by three dioptrics inseparably connected with the structure of the medium—or, better still, B, & by an infinitely thin ideal tube led through the ponderable particles. The refracted light will pass through the tube without striking against its sides, even when the medium is moved in space with any velocity of translation whatever. And yet, in consequence of the motion, the angle of incidence, angle of refraction, and ratio of refraction change. 2 H 2 420 Prof. E. Ketteler on the Dispersion of Light or cos 6 cos? a 4 cos”? b _ cos” ¢ : eS a SSS Y ie rAd + a) r(1 + B) 7 (1 +y¥) ‘ : i (21) a Pie .COS7 a cos? b cos? ¢ iene Cue) If the penultimate of these be introduced into equation (18), and at the same time we put for shortness sce NE a (5) aie Deel ge @) r (lea) Neg 7, 8) Ney aes the equation takes the following definitive form :— m lat (5) cos? a+ (5) co: b+ (5) cos? eh. (22) 1 If, finally, we epitomize the three sums as A, B,.C, it is written more briefly, n= (1+ A) cos?a+(14+B) cos?b+ (1+ C) cos’ e, |: 1 cos*a cos*b cos*e (23) ae 2 2 3 2 Oo oO ®> QO, and in this form it may have henceforth to replace the “second ’”’ or Pliicker’s ellipsoid (€), the expression of which has hitherto been = = 1 Oo With the ellipsoid represented by equation (23) a second is then associated, represented by Le COs ta, RCS: be meeOss cA no TA Oa ee a. Dik, Boye Byer? BAe @ =) COS” a, + @3 Cos” by + 2 Cos” Cp. This may take the place of what has hitherto been named the first or reciprocal ellipsoid (#1), for which it has up to the pre- sent been assumed that 7 cos6=o. With the two planes mentioned the further theory of double refraction is, as is known, completely traced. 10. Up to this point we have held fast to the special case for compound media, that the angle between the ray and the wave-normal has the same value for all the individual optico- chemical elements ; we can now drop this supposition. Even with the most common composition of the medium, the oscil- latory motion about a point of it will, after the lapse of the time-unit, have advanced as far as to a perfectly determined closed plane. ‘To a determinate radius vector of the same cor- in Singly and Doubly Refracting Media. 421 responds a determinate compatibility on the part of the zethe- real and corporeal particles; and the resistance called forth by the presence of the latter now comes into play as a partial com- onent (€; COs 6,, %€cose,) in a direction (a), b4, cy; a’, 6”, ¢’,) depending on the effective molecular quality. Here, therefore, afterwards as before, 6 denotes the angle be- tween a definite compound series of molecules a, b, c, resulting from the modification, and a simple constituent thereof in its unmodified position. Hence we may, and must, attribute to one and the same ray any number of partial normals and ex- citing partial waves, all of which combine into a resultant normal and wave; and this resultant is obtained when we erect on the corresponding radius of our direct ellipsoid a tan- gential plane and let fall a perpendicular upon it. The plane determined by the radius vector and the normal is then the resultant vibration-plane ; and the angle A between them becomes the resultant angle between the ray and the resultant normal ; so that we get OG COS Nani eee: tyra tease (29) If now the various kinds of mass-particles are modified by the partial pressures applied in identical axial directions, or rather, when we introduce corresponding molecular forces, if the structure constituted by the individual heterogeneous ele- ments is arranged symmetrically about the same directions, this corresponds to the case of the regular system of crystals ; but does it consist of a grouping about divergent axial direc- tions, then we have what is called the dispersion of the optical axes, which has hitherto seemed to mock all attempts to ex- plain it. 11. Let us now, for anisotropic media also, turn from the moving forces to the vires vive. For this purpose let us con- struct the plane of vibration corresponding to a determined colour and direction of the ray, and in it a parallelogram L MNO, making its longer side OL parallel to the ray and equal to Ul’, giving to the shorter side ON the length (A=A, cos A) of the actual amplitude and making it coincide with the direction of vibration of the ether particles, so that consequently it will be perpendicular to the resultant normal and the angle LON be equal to 90°+ A. Let us further imagine the ethereal and corporeal particles situated within it brought out of their position of equilibrium into an extreme position such as would correspond to a wave characterized by A, /’, and somehow kept fixed therein. The elasticity thereby accumulated is again the same as if the cor- poreal particles were not present. If the medium be then left to itself, the ether particles will press back in oblique paths of 422 On the Dispersion of Light in Refracting Media. the length A, parallel to N O, towards the position of equili- brium, and carry with them the corporeal particles in paths the equivalent length of which will be A’. If for the position of equilibrium all the elasticity is transformed into vis viva, 2 this resolves itself as before into mvs + >m/ 7" If, on the other hand, in pure ether the same initial dis- placement is produced in order to attain the same elasticity, we have, in order that we may replace the interfering external force by a wave-motion of the same amplitude A and the same deviation, to bring the above parallelogram, by twisting its sides, into the form of a rectangle, and also to shorten its length /’ to /, while 7must be made =/’cosA. The maximum vis viva corresponding to this displacement is ae slag SNe 2 mp8 = Mpg = pa The same end, however, is attained by displacing the zther particles A,, while retaining the same extent /’, and forming a rectangle out of A, and /’ instead of A and /. It is pre- cisely 2 Dg = A®. Accordingly we