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DIELECTRICS AS AFFECTED BY TEMPERATURE AND TIME. 149
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The wires were again connected and the circuit left closed for
about twenty-four hours. It was found that the plug had ex-
tended for about half an inch on the side of the sulphuric acid by
the formation of crystals of sulphate of potash, but had not
apparently changed where it was in contact with the potash.
X. A similar plug was formed in a second tube. Into one
limb sulphuric acid, with a small quantity of permanganate of
potash, was poured, into the other caustic potash : E. F. = 178.
Circuit was closed for ten minutes.
E.F
10 seconds after insulation,
90
20
no
40 • „
123
1 minute „
128
3 minutes „
138
9
148
19
150
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45.
ON THE QUASI-RIGIDITY OF A RAPIDLY MOVING
CHAIN.
[From the Proceedings of the Birmingham Philosophical Society, "l
Read May 9, 1878.
As diagrams would be necessary to an intelligible description
of the apparatus employed or of the detail of the phenomena
exhibited, it appears well to confine this abstract to a statement
and short explanation of the more general dynamical properties of
a moving chain, the more so as the experiments are very ftilly
described by the inventor of most of them, Mr Aitkin.
Briefly, the apparatus consists of an endless chain hanging in
a loop over a pulley which could be caused to revolve about a
horizontal axis, so giving a rapid motion to the chain. It is firstly
observed that the motion of the chain does not very materially
affect the form in which the chain hangs when it attains equi-
librium or a state of steady motion. The chain being at rest
its form is a catenary : what forces must be applied to each small
portion of the chain to keep the form the same when it is in
motion? Any such small portion is at any point moving, with
velocity (F) the same for all points of the chain; hence, if R be
the radius . of the circle most nearly agreeing with the chain at
the point (the circle of curvature), it follows that the change of
motion is towards the centre of this circle at a rate -^ . Now it
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348 ON THE QUASI-RIGIDITY OF A RAPIDLY MOVING CHAIN.
is also easy to show that a tension (T) in the chain will give a
resolved force towards the centre on an element of chain, length
Tda
dsy mass mdSy equal to -p- . If, then, the tension of the chain be
increased beyond that due to the forces acting upon it when at
rest by the amount mV\ constant for all parts of the chain and
quite independent of iJ, this will be precisely sufl&cient to effect
the actual changes of motion when the velocity is F, and the
effect of such velocity will be not to alter the form but merely the
tension of the chain.
Consider now a straight chain, stretched with tension T, Let
the chain be struck at any point; two waves will be caused
/T
travelling in opposite directions with velocity a/ — . The height
of these waves will be greater as the blow is greater, and less as
the tension is greater ; in fact, the height of the wave will vary
directly as the blow, and inversely as the velocity of transmission
of a wave. Suppose, now, the stretched chain be caused to
move along its length with velocity F, = a/ — , how will these
waves appear ? That which is moving in a direction opposite to
the motion of the chain will appear stationary to the observer as a
rumple on the chain, whilst the other will appear to move away
with velocity 2F. It will also appear to such observer that to
produce a rumple of given height he must strike a greater blow as
the velocity and tension of the chain are greater : that is, if the
velocity of the chain be doubled he must either strike twice as
hard or strike two blows of the same value ; or, if he be applying a
continuous force to raise the rumple, he must apply it either twice
as hard or twice as long. Let now the moving chain be curved,
not straight ; any small length of it may be regarded as sensibly
straight, and we may conclude that the effect of any very small
blow will be the same as if all the chain were straight, thus far,
that it will cause a rumple fixed relatively to the observer, of
which the height is inversely proportional to the velocity F, and a
wave which will run away at a velocity 2 F
We may now further explain the observations. When the
chain is hanging in a catenary and in rapid motion, strike it
a blow. As we should expect from the foregoing reasoning, the
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ON THE QUASI-RIGIDITY OF A RAPIDLY MOVING CHAIN. 349
effect is different on a moving chain and on one at rest. The
chain presents a sort of rigidity greater as the velocity is greater ;
the blow causes a rumple or dint, which would remain firm in
position but for the action of gravity. Suppose the blow to be
struck at the ascending side of the loop, two effects are observed.
The rumple just mentioned travels downward with decreasing
velocity till it reaches the bottom of the loop, where it remains as
an almost permanent deformation; but besides this, sensibly at
the instant when the blow is struck, a second rumple appears on
the chain at the point where it meets the pulley, and travels
downwards like the first with continually diminishing velocity.
The explanation is easy : — As already shown the tension of the
chain at any point consists of two parts, that due to the weight of
the chain below and that due to the velocity of the chain, — the
velocity of the chain is then less than that corresponding to the
tension. If we strike the chain we shall have two waves produced,
one not quite stationary, but travelling slowly in a direction
opposite to the motion of the chain, and stopping when it reaches
the bottom of the loop, where the tension of the chain at rest is
sensibly nil ; the other, running up with a velocity a little more
than double that of the chain, is reflected at the pulley, and then
travels slowly downwards like the first.
The above will suggest the explanation of many other experi-
ments. We will here only deal with one as a further example.
The chain is kept in contact with one point of the pulley by means
of a second pulley, pressed by the hand against it in a horizontal
direction at the point where it comes in contact with the first
pulley on the ascending side ; a piece of board is brought into
contact with the lowest point of the loop of chain and somewhat
rapidly raised — the chain stands up upon the board like a hoop of
wire, rising up from the pulley to a height of perhaps three or
four feet above it. The pressure of the board in the first instance
diminishes the tension of the chain at its lowest point. This
diminution will instantly extend throughout the chain, and may
render the tension even at the highest point of the chain less than
that due to the velocity. If that be so, that highest point will
recede from the centre about which it is moving — that is, will rise
from the pulley.
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46.
ON THE TORSIONAL STRAIN WHICH REMAINS IN A
GLASS FIBRE AFTER RELEASE FROM TWISTING
STRESS.
[From the Proceedings of the Royal Society, No. 191, 1878.]
Received October 4, 1878.
It has long been known that if a wire of metal or fibre of glass
be for a time twisted, and be then released, it will not at once
return to its initial position, but will exhibit a gradually decreasing
torsion in the direction of the impressed twist. The subject has
undergone a good deal of investigation, especially in Germany.
The best method of approximating to an expression of the facts
has been given by Boltzmann (Akad, der Wissensch. Wien, 1874).
He rests his theory upon the assumption that a stress acting for a
short time will leave after it has ceased a strain which decreases
in amount as time elapses, and that the principle of superposition
is applicable to these strains, that is to say, that we may add
the after- eflfects of stresses, whether simultaneous or successive.
Boltzmann also finds that, if (t)T be the strain at time t
resulting from a twist lasting a very short time t, at time ^ = 0,
A
(^) = -— , where A is constant for moderate values of t, but
V
decreases when t is very large or very small. A year ago I made
a few experiments on a glass fibre which showed a deviation from
Boltzmann's law. A paper on this subject by Kohlrausch (Pogg,
Ann,, 1876) suggested using the results of these experiments to
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TORSIONAL STRAIN WHICH REMAINS IN A GLASS FIBRE, ETC. 351
examine how Boltzmann*s law must be modified to express them.
Professor Kohlrausch*s results indicate that in the cases of silver
wire and of fibre of caoutchouc Boltziinann's principle of super-
position is only approximate, and that in the case of a short
A
duration of twisting 0(^)=— , where a is less than unity; in case
If
of a long duration of twisting he uses other formulae, which pretty
successfully express his results, owing in part no doubt to the
fact that in most cases each determination of the constants applies
only to the results of one duration of twisting. In a case like the
present it appears best to adopt a simple form involving constants
for the material only, and then see in what way it fails to express
the varying conditions of experiment. In 1865 Sir W. Thomson
published (Proceedings of the Royal Society) the results of some
experiments on the viscosity of metals, the method being to
determine the rate at which the amplitude of torsional vibrations
subsided. One of the results was that if the wire were kept
vibrating for some time it exhibited much greater viscosity than
when it had long been quiescent. This should guard us from
expecting to attain great uniformity in experiments so roughly
conducted as those of the present paper.
2. The glass fibre examined was about 20 inches in length.
Its diameter, which might vary somewhat fix)m point to point, was
not measured. The glass from which it was drawn was composed
of silica, soda, and lime ; in fact, was glass No. 1 of my paper on
" Residual Charge of the Leyden Jar" (Phil. Trans,, 1877). In all
cases the twist given was one complete revolution. The deflection
at any time was determined by the position on a scale of the
image of a wire before a lamp, formed by reflection from a light
concave mirror, as in Sir W. Thomson s galvanometers and quad-
rant electrometer. The extremities of the fibre were held in
clamps of cork ; in the first attempts the upper clamp was not
disturbed during the experiment, and the upper extremity of the
fibre was assumed to be fixed ; the mirror also was attached to the
lower clamp. This arrangement was unsatisfactory, as one could
not be certain that a part of the observed after-eflfect was not due
to the fibre twisting within the clamps and then sticking. The
diflSculty was easily avoided by employing two mirrors, each
cemented at a single point to the glass fibre itself, one just below
the upper clamp, the other just above the lower clamp. The
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352 ON THE TORSIONAL STRAIN WHICH REMAINS IN A
upper mirror merely served by means of a subsidiary lamp and
scale to bring back the part of the fibre to which it was attached
to its initial position. The motion of the lower clamp was damped
by attaching to it a vane dipping into a vessel of oil. The
temperature of the room when the experiments were tried ranged
from 13° C. to IS'S^'C, and for the present purpose may be regarded
as constant. The lower or reading scale had forty divisions to the
inch, and was distant from the glass fibre and mirror 38| inches,
excepting in Experiment V, when it was at 37^ inches. Sufficient
time elapsed between the experiments to allow all sign of change
due to after-effect of torsion to disappear. In all cases the first
line of the table gives the time in minutes from release from
torsion, the second the deflection of the image from its initial
position in scale divisions.
Experiment I. — The twisting lasted 1 minute.
t 1 2 3 4 5 7 10 17 25
Scale divisions... 22 13 9 7 5^ 4 3 2 1
Experiment II. — The twisting lasted 2 minutes.
t 1 2 3 4 5 7 10 20 40
Scale divisions... 38 25 18 15 13 10 8 4^ 3^^
Experiment III. — Twisted for 5 minutes.
t 12 3 4 5 7
Scale divisions... 64 51 41^ 35^ 32 26^
t 10 15 22 58 15
Scale divisions... 21^ 17 14 7 2
Experiment IV. — Twisted for 10 minutes.
t J 1 2 3 4 7 10
Scale divisions... 106 85 66 57 49^ 37^ 31
t 15 25 45 120 170
Scale divisions... 24^ 18 13 7 6
Experiment V. — Twisted for 20 minutes.
t 1 2 3 4 5 7 10
Scale divisions... 110 89 75 68 61^ 52 44
t 15 25 40 60 80 100
Scale divisions... 35^ 26^ 21 18 13^ 12 J
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— - ^_^^ ^lm r. ■ r, rf.Hi^ ■
GLASS FIBRE AFTER RELEASE FROM TWISTING STRESS. 353
Experiment VI. — Twisted for 121 minutes.
t i 1 2 3 4 5 7
Scale divisions... 191 170 148 136 126^ 119^ 108^
t 10 15 30 65 90 120 589
Scale divisions... 97 84^ 63^ 41^ 34 28 3^
It should be mentioned that the operations of putting on the
twist and of releasing each occupied about two seconds, and were
performed half in the second before the beginning and end respec-
tively of the period of twisting, and half in the second after or as
nearly so as could be managed. The time was taken by ear from
a clock beating seconds very distinctly.
3. The first point to be ascertained from these results is
whether or not the principle of superposition, assumed by Boltz-
mann, holds for torsions of the magnitude here used.
If the fibre be twisted for time T through angle X, then the
torsion at time t after release will be X {y^{T -\- ~'^(0)i where
If now 7 = ^1 + ^2 + ^ + ... we may express the etfect of one
long twist in terms of several shorter twists by simply noticing
that
Z{t(0->/r(«+!r)} = Z[(>|r(0->|r(^4-^)}
Apply this to the preceding results, calculating each experi-
ment from its predecessor. Let Xt be the value oi y^ {T •\- 1) ^ -^ {t\
that is, the torsion at time ty when free, divided by the impressed
twist measured in same unit ; we obtain the following five tables
of comparison.
Results for r= 2 compared with those from T^\,
t 1 2 3 4 5 7
a?e observed 000195 128 092 077 066 051
a^e calculated 0-00199 112 082 064 051 040
t 10 20 40
Xt observed 041 023 018
Xt calculated 029 016
H. II. 23
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354 ON THE TORSIONAL STRAIN WHICH REMAINS IN A
Results for r= 5 compared with those from 7=2 and 7=1.
t 1 2 3 4 6 7 10
art observed 0-00328 262 212 182 164 136 110
art calculated 000323 233 181 156 136 108 193
t 15 22 58 151
a?t observed 087 072 036 010
xt calculated 066 047
Results for T = 10 compared with those from T— 5.
t 4 1 2 3 4 7 10
art observed 000544 435 338 292 253 192 159
art calculated — 469 398 339 eSOO 236 197
t 15 25 45 120 170
art observed 125 092 067 036 031
Xt calculated 161 130 088
Results for 7= 20 compared with those from T= 10.
t 1 2 3 4 5 7 10
art observed 000580 470 398 358 327 276 234
art calculated 000587 483 430 384 356 312 266
t 15 25 40 60 80 100
art observed 188 140 111 085 072 066
art calculated 217 167 135 100 084
Results for 7= 121 compared with those from T= 20.
t
i
1
2
3
4
5
7
Xt observed
000979
871
758
697
648
612
556
Xt calculated
—
1070
950
880
830
780
730
t
10
15
30
65
90
120
589
Xt observed
497
433
325
212
174
144
18
Xt calculated
670
600
500
380
350
In examining these results it must be remembered that those
for small values of T are much less accurate than when T is
greater, for the quantity observed is smaller but is subject to the
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GLASS FIBRE AFTER RELEASE FROM TWISTING STRESS. 355
same absolute error; any irregularity in putting on or releasing
from the stress will cause an error which is a material proportion
of the observed deflection. For this reason it would be unsafe to
base a conclusion on the experiments with T=l and r=2. The
three last tables agree in indicating a large deviation from the
principle of superposition, the actual effect being less than the
sum of the separate effects of the periods of stress into which the
actual period may be broken up. Kohlrausch finds the same to be
the case for india-rubber, either greater torsions or longer dura-
tions give less after-effects than would be expected from smaller
torsions and shorter periods.
A
4. Assuming with Boltzmann that (^) = — , we have at time
t
t after termination of a twist lasting time T,
Xt^A {log(r4-0-log^},
the logarithms being taken to any base we please. The results
T + 1
were plotted on paper, Xt being the ordinate and log — - — the
t
abscissa ; if the law be true we should find the points all lying on
a straight line through the origin. For each value for T they do
lie on straight lines very nearly for moderate values of t ; but if T
is not small these lines pass above the origin. When t becomes
large the points drop below the straight line in a curve making
towards the origin. This deviation appears to indicate the form
A
(^) = — , a being less than, but near to, unity. If a = 0*95 we
z
have a fairly satisfactory formula :
a;t = A'(Tni^-f\ where ul' = ,-^ when ^=121.
In the following Table the observed and calculated values of a)t
when r= 121 are compared, A' being taken as 0032.
t
4
1
2
3
4
5
7
xt observed
0-00979
871
758
697
648
612
556
Xt calculated
000976
870
755
691
643
600
550
t
10
15
30
65
90
120
589
Xt observed
497
433
325
212
174
144
18
Xt calculated
493
429
320
218
176
147
42
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t
15
Xt observed
188
Xi calculated
185
356 TORSIONAL STRAIN WHICH REMAINS IN A GLASS FIBRE, ETC.
To show the fact that A' decreases as T increases if a be
assumed constant, I add a comparison when T= 20, it being then
necessary to take -4' = 0'037.
t 1 2 3 4 5 7 10
are observed 000580 470 398 358 327 276 234
art calculated 000607 485 422 370 337 285 233
25 40 60 80 100
1*0 111 085 072 066
125 089 067 052 041
A better result would in this case be obtained by assuming
a =092, or = 093 in the former case with ^' = 0021. Probably
the best result would be given by taking A constant, and assuming
that a increases with T.
A
Taking the formula <^(f)= — these experiments give values of
V
A ranging from 00017 to 0*0022. Boltzmann for a fibre, probably
of a quite diflferent composition, gives numbers from which it
follows that ul= 0-0036.
5. In my paper on "Residual Charge of the Leyden Jar*'* that
subject is discussed in the same manner as Boltzmann discusses
the after-effect of torsion on a fibre, and it is worth remarking
that the results of my experiments can be roughly expressed by a
A
formula in which (t) = zi. For glass No. 5 (soft crown) a = 065,
z
whilst for No. 7 (light flint) it is greater; but in the electrical
experiment no sign of a definite deviation from the law of super-
position was detected.
• Supraf p. 19.
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47.
ON THE STRESSES CAUSED IN AN ELASTIC SOLID
BY INEQUALITIES OF TEMPERATURE*
[From the Messenger of Mathematics^ New Series, No. 95,
March, 1879.]
Various phenomena due to the stresses caused by inequalities
of temperature will occur to everyone. Glass vessels crack when
they are suddenly and unequally heated, or when in manufacture
they have been allowed to cool so as to be in a state of stress when
cold. Optical glass is doubly refracting when badly annealed or
when different parts of the mass are at different temperatures.
Iron castings which have been withdrawn from the mould whilst
still very hot, or of which the form is such that some parts cool
more rapidly than others, are liable to break without the applica-
tion of any considerable external stress. The ordinary theory of
elastic solids may easily be applied to some such cases.
Let M, Vy w be the displacements of any point {xyz) of a body
density p, parallel to the coordinate axes. Let N^, N^^ N^y Ti, T^y
Tz be the elements of stress; i.e. NiU is the tension across an
elementary area a resolved parallel to x, the element a being
perpendicular to x\ 2\^ is the shearing force across an element fi
resolved parallel to Zy yS being perpendicular to y ; T^ is then also
the shear parallel to y across an element perpendicular to z,
* See Report of the British Association for 1872, p. 51.
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358 ON THE STRESSES CAUSED IN AN
If pX, pTf pZ be the external forces at {xyz)
dx dy dz
dx dy dz '^
} (!)•
dx dy dz '^
These are strictly accurate. Of an inferior order of accuracy
are the equations expressing the stresses in terms of the strains of
an isotropic solid
■KT ^ /% cs du ^
'•-(s-SJ
.(2),
dy)
du dv dw ,1.1. • mi
where ^=^-+^-4--^ = the dilatation at the point. These
dx dy dz ^
equations are inaccurate, inasmuch as they are inapplicable if
the strains be not very small, and as even then in all solids
which have been examined the stresses depend not only on the
then existing strains but in some degree on the strains which
the body has suflfered in all preceding time (see Boltzmann, Akad.
der Wissensch. zu Wien, 1874 ; Kohlrausch, Pogg. Annalen, 1876 ;
Thomson, Proceedings of Royal Society, 1865 ; some experiments
of my own, Proceedings of Royal Society, 1878*; Viscosity in
Maxwell's Heat),
Assuming equations (2) we observe that as these and also (1)
are linear, we may superpose the eflFects of separate causes of
stress in a solid when they act simultaneously.
Equations (2) are intended to apply only to cases in which
when the stresses vanish the strains vanish, and in which the
strains result from stress only and not from inequalities of tem-
perature. The first limitation is easily removed by the principle
of superposition. We must determine separately the stresses
when no external forces are applied, and then the stresses due to
the external forces on the assumption that the solid is unstrained
when free and finally add the results. For example, if we are
considering a gun or press cylinder, we know that internal
* Supray p. 360.
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ELASTIC SOLID BY INEQUALITIES OF TEMPERATURE. 359
pressure will produce the greatest tension in the inner shells, and
we can hence at once infer that if the gun or cylinder be so made
that normally the inner shells are in compression and the outer
in tension it will be stronger to resist internal pressure.
To ascertain the effect of unequal heating, assume that \, fi are
independent of the temperature, an assumption of the same order
of accuracy as assuming in the theory of conduction of heat that
the conductivity is a constant independent of the temperature.
Let K be the coefficient of linear expansion, t the temperature
at any point in excess of a standard temperature. If there be no
stresses,
du _ dv dw _
dx'~ dy^ dz " '
therefore
du
{S\ + 2fi)KT=\0+2fi
dx'
"-"(IM).*-
if there were stresses, but t were zero,
iV', = \^ + 2/i^,&c.;
superposing effects we have
du
N,^\0+2fi^- (3\ + 2fi) KT
dw dv\
rr, (dw dv\
dy
Substitute in the equations of equilibrium
where
de
dr
(X + ^)-^ + ;.V»«-7;j-+pX =
dx
dx
7 = (3\ + 2/1*) K.
.(3).
.(4),
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360 ON THE STRESSES CAUSED IN AN
If there be equilibrium of temperature VV = 0, and the effect
of unequal heating is exactly the same as that of an external force
potential ^ — ; in this case we have the equations
'■^-'Sj (5).
still true and under the same conditions.
Examine the case when there are no bodily forces and when
everything is symmetrical about a centre. The displacement at
any point is radial, call it Uy and the principal stresses are radial
and tangential, call them 12 and T.
The equation of equilibrium is
dr
and the stresses are expressed by
-2rT=0 (6),
T=\0'^2fJL- -7T
r* dr
.(7);
substituting
therefore
tx^9 ^{dnr^^idu ^u\ dr ,^,
r^U=--^JrHdr + a.r^ + b (9),
\ "T" ^JUL
where a and 6 are constants to be determined by a knowledge of
R OT U for two specified values of r. This equation is of course
true whether there be equilibrium of temperature or not.
The interior and exterior surfaces of a homogeneous spherical
shell are maintained at different temperatures, to find the resulting
stresses.
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ELASTIC SOLID BY INEQUALITIES OF TEMPERATURE. 361
Let 7*1, r, be the internal and external radii, ^i, ^ the internal
and external temperatures, then if t be the temperature at
radius r,
/
T= C+-
where
and
c =
r
y
Substitute in equation (9) and then in (7)
.(10).
lr=
Gcr+i/) + ar + -
X + 2/i
■ ^= — ^ fc + '^) + 3a = ;-^„- T +
\ + 2/t\ r I X + 2/*
3a
)■
..(11),
R^.^Jm^.
3(x.2,)^ x¥|l^(^^-^-)«-^;
write R in the form
where
^^ 2/LC7 { U-ti)r{r^
for we shall not require to find CT; we find
whence
^ - ''^^Zg ""•■ {- (- -^ -) 4- -^±1^ - rffi] (12).
iJ will have a maximum or minimum value when
r» =
SnVa^
^2* + ^1^2 + n*
and its value then is
raVS
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362 ON THE STRESSES CAUSED IN AN
this is positive if ^>^, as we may see at once from physical
reasons.
Now
^'Tr-dr -^1-(^» + ^0 + ^ + -2^(,
if U>ti; this decreases as r increases; when r — ri, its value
becomes
R/ | -2(r, + ri)7-i + (r,' + rir, + r,») + r^^)
^1 -2^^. 1
The case when the thickness is small is interesting. Let
^2 = ^1 + a:, then the maximum tension is
neglecting the term — in comparison with unity, we see that of
^1
two vessels the thicker is not sensibly more liable to break than
the thinner, a result at first sight contradictory to experience.
The explanation is that the greater liability of thick vessels to
break is due to the fact that, allowing heat to pass through but
slowly, a greater difference of temperature between the two
surfaces really exists.
Let t^\ ti' be the actual surface temperatures, we may assume
that, if ^2 and ^i be the temperatures of the surrounding media, the
heat passing the two surfaces per unit of area will be Hiit^—U)
andiri(^'-^).
Hence
using this in the equation last obtained we have a result quite in
accord with experience.
* This result was set by me in the recent Mathematical Tripos Examination
(Friday afternoon, January 17, 1879, Question ix).
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ELASTIC SOLID BY INEQUALITIES OF TEMPERATURE. 363
Returning now to equations (7) and (9), suppose the sphere to
be solid and to be heated in any manner sjonmetrical about the
centre. The constant b must vanish, and
i2 = (X+2/i)^-4/i 7T
= -4/i— + 3(XH-2/A)a
Now the mean temperature within the radius r is
^irjr^rdr _ Sfr^rdr
therefore, since the pressure is zero at the surface of the sphere,
iZ = ^ . o \ - {^^an temperature of whole sphere — mean tem-
perature of sphere of radius r] (14),
'■-i?^ c^).
= ^ , > . {mean temperature of whole sphere — f t + J mean
temperature within the sphere of radius r] (16).
Other problems of the same . character as the preceding will
suggest themselves, for example that of a cylinder heated sym-
metrically around an axis, but as no present use could be made of
the results I do not discuss them.
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48.
ON THE THERMO-ELASTIC PROPERTIES OF SOLIDS.
(Published in 1879 as an Appendix to Clausius*
" Theory of Heat*!')
Sir William Thomson was the first who examined the thermo-
elastic properties of elastic solids. Instead of abstracting his
investigation {Quarterly Mathematical Journal, 1855) it may be
well to present the subject as an illustration of the method of
treatment by the Adiabatic Function.
Consider any homogeneously-strained elastic solid. To define
the state of the body as to strain six quantities must be specified,
say u, Vy Wy x^yyZ: these are generally the extensions along three
rectangular axes, and the shearing strains about them, each
relative to a defined standard temperature and a state when the
body is free from stress. The work done by external forces when
the strains change by small variations may always be expressed in
the form
{Uhu + Viv + . ..) X volume of the solid,
because the conditions of strain are homogeneous. CT, V. are
the stresses in the solid : each is a function of u,v and of the
temperature, and is determined when these are known. Let
denote the temperature (where ^ is to be regarded merely as the
name of a temperature, and the question of how temperatures are
to be measured is not prejudged).
Amongst other conditions under which the strains of the body-
may be varied, there are two which we must consider. First,
suppose that the temperature is maintained constant ; or that the
* Translation by W. B. Browne, M.A.; Macmillan and Co., p. 363.
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ON THE THERMO-ELASTIC PROPERTIES OF SOLIDS. 365
change is efifected isothermally. Then 6 is constant. Secondly,
suppose that the variation is efifected under such conditions that
no heat is allowed to pass into or to leave the body ; or that the
change is efifected adiabatically. In the latter case 0, u,v,
are connected by a relation involving a parameter which is always
constant when heat does not pass into or out of the solid : this
parameter is called the adiabatic function.
We have now fourteen quantities relating to the body, viz. six
elements of strain, six of stress, the quantity which defines the
temperature, and the parameter (f> the constancy of which imposes
the adiabatic condition. Any seven of these may be chosen as
independent variables.
Let the body now undergo Camot's four operations as
follows : —
1®. Let the stresses and strains vary slightly under the sole
condition that the temperature does not change. Let the conse-
quent increase of (f> be B*