OF HEAT IN' ELLIPSOIDS OF REVOLUTION. 
123 
and finally, 
c/E = c 3 sinh a sin /3(cosh° a — cos 2 /3)d(fx^ad/3 
(14) 
5 . We proceed to find the solution of ( 12 ) which is appropriate. We may satisfy 
it by putting Y=2(cos ?rui>U l + sin m<t> XL), where Uj and U c both satisfy 
d 2 U d 2 U 
dY + d+ 
dTJ 
dU 
+ c °th a^+cot {3 — -m 2 ( 
*1 1 
, sinh 2 
+Yi) u =f ( cosh2 
« sin- j3J f 
a —cos 3 /3) 
dU 
dt' 
And with regard to m, it must be observed that it cannot be other than a whole 
o 1 
number, since the value of Y must repeat itself in going round the surface of the 
ellipsoid in the ^-direction ; that is to say, 
We may also put 
771 = 0, 1, 2 ... co 
(15) 
U=e k ' n .v 
v=3j(P).nJ( 
(16) 
where 3 and H are functions of /3 alone and of a alone respectively, determined from 
the equations 
g+ c ot/3‘|-^>=\%W/»-B.(18,) 
d-9 , d9 m~ . 0 , 0 , . 
- 7-5 +coth a— - . , n = — \~c 2 cosh- a9.-\-k9 . (19) 
do? d« sinh 2 « 1 v 
wherein k is a constant, as yet undetermined. In the sequel it will appear that k 
has an infinite number of values for a given value of m and a given value of X, and one 
of the objects of the present investigation is to furnish the equation which determines 
it, and to approximate to its different values when the eccentricity of the ellipsoid is 
small. In this respect the present problem differs essentially from the corresponding 
one for a sphere in which k is indejDendent of X ; it is then given by 
k—n[n + 1 ), where n=m, m+ 1 , 777 + 2 , . . . 00 . 
If we put cos/3 —v, the equation in 3- may be written 
(l-* 2 )ff-2 v~^- n 3=XW3-k3 . . . . 
dv~ <iv 1 — v~ 
(17) 
and, if we write %—Xc sinh a, the equation in 9. is 
e 2 
(18) 
