OF HEAT IN ELLIPSOIDS OF REVOLUTION. 
149 
These are the equations which determine the different values of A. and the corres¬ 
ponding-ratios C 0 : C\: Ch: . . . . When we neglect powers of e 3 beyond e 4 ', the equation 
determining \ is 
1 — 22 
oo l dfl r 
g/g/ an. 
dz 
+ffi 
>0 
df>s 
df 
+g;n. 
, r not=s; 
■ • (71) 
If the first power of e 2 alone is retained, this equation breaks up into 
jj+g/n, = o, —'+ 0 /n,=o, &c. (72) 
The solutions of these equations may be easily found from the corresponding results 
for the sphere, and the quantities g/ . . . have been already found. 
19. We shall not pursue this inquiry further, but shall now show how the arbitrary 
constants introduced into the solution may be determined. As already explained in 
Art. 2 of this paper, when the solution is represented by V=2Ae~ A2f A, A is to be 
found from 
a[^e=[w 0 ^e, 
and what we have to find is, therefore, 
/"Sir r+l ra„ ^ _ 
I v 2 sinh a cos /3 (cosh 3 a — cos 2 /3)d<f)df3da. . . . 
• o--no 
(73) 
If the surface is kept at a constant temperature zero, or if we adopt the simple law 
dV , A 
of radiation ^--b W=0, we may take 
r=cos 
as the type of solution : the above integral may then be resolved into the components 
r +i r+i i r“o # C “o . n 
3-. &dv, v 2 9-. S-dv, sinh afl. f Idct. sinh a. cosh 3 afl. fid a ; and of these the two 
J -i -’-l Jo Jo 
former have been already found. 
For the accurate solution of the problem of radiation we must take for v an expres¬ 
sion of the form 
cos wi^)(c 0 ^ 0 fl 0 +c 1 T 1 n 1 -hc 2 A 2 n 3 + . . . ), 
in which c 0 c l . . . are known constants, one of which is to be put equal to unity. 
When we substitute this expression for v we perceive that we have still another 
integral to evaluate, namely, 
sinh afl. fl'da, 
Jo 
