OF CRYSTALS AND OTHER BAD CONDUCTORS 
497 
where cosh ax x , sinh ax x , cosh ax 2 , sinh ax 2 are known quantities, to determine the 
values of A and B. 
Differentiating the equation v = A cosh ax + B sinh ax and putting a — 0, we have 
at the surface in contact with the disc, v = A, dv/dx = aB. 
The isothermal surfaces in the discs themselves may be assumed to be planes, for a 
small calculation, like that ma.de (p. 490) for the bar itself, shows that the radius of 
curvature of these surfaces is about 40 cms. Writing ct x , a 2 for the coordinates of the 
surfaces of the disc, A x , A 3 the temperatures determined as above, Q t , Q 2 the values of 
ZqaB in the bars at the surfaces in contact with the disc, k x being the conductivity of 
bars, and k that of the disc, we have, for the temperature in the disc itself: 
Ao sinh • x - a x + A x sinh . a„ - 
qk 
X 
sinh P 4ct 2 - a x 
Differentiating and writing down the expressions for the flow of heat into and out 
of the disc we have the equations— 
, , Ajj-Ajcosh \/ 'k «a ~ «i 
Q* = # V-'-—- 
qk 
and 
QafZi — Qk \J 
7 Ao cosh 
ph * 
qk 
sinh 
V 
pll 
qk 
vli - 
— — a, 
qk 1 
A: 
sinh \/ ^ ‘ % - 
where p, h, q, k refer to the disc and have the usual meanings, q being = q x in most 
cases, differing only slightly in others. Now a 3 — a l5 the thickness of the discs, is 
small enough to make /\J'a 2 — a x small. Hence, writing a 2 — a x = t, and ex¬ 
panding the hyperbolic functions, we have, as a close approximation— 
QiQi = qk \/\ 
Therefore 
W,(.+g-f 
■VS ■(■+'-- ?v 
'ph 
qk 6 
k = (& 
2/ 
Ao — Aj (1 -f- 
similarly 
k = 
ph t~ 
qk ’ 2 
3 s 
MDCCCXCII.—A. 
