OF HEAT IN ELLIPSOIDS OF REVOLUTION. 129 
If we now turn to equation (19), and compare the result just obtained with (20), 
we see that we can satisfy the equation (19) by 
H —h'i(cto • • • ).(31,) 
Where a 0 , cq, cc. 2 . . . are connected by the same relations as before—namely, I. of 
(22), and the values of k are still, as before, the roots of c^^O. 
It will be observed that this result depends on the identity of the forms of the 
right-hand members of (31) and (20). But the transformation (30) depends essentially 
on the hypothesis that n—m is even. For the values of k which depend on n—m odd, 
and give rise to the second class of expressions for d, this method fails completely; 
in other words, distributions of heat which are not symmetrical about the equator of 
the ellipsoid cannot be represented by H-functions of the type we have just found. 
I was led however to expect, from other expressions which will be given presently, that 
the true form in this case was to be discovered by putting 
.( 32 ) 
After substitution and reductions, we obtain 
d 2 fl' d ft' 
a~u 
d f 
r , i —i 
TrfF +n+ r 
Now the value of the expression 
dr S„ 1 dS„ 0 1— mr a _ 3 dS„ n(n + 1) +1 — m 2 a 
by (29) 
(n+2) 2 — m 2 q , 2n* + 2n — 2m 2 -1 0 , (n — l) 2 — m 2 C1 
— (2n+l)(2n + 3) CVfS ‘ 1_ (2n-l)(2n + 3) ^~ 1 ~(2rc-l)(2w+lj "" 3 
If we put 
«_ 1.3.5 . . . (2n-l) _ 
H (n 2 — m 2 )(n—2 2 —m 2 ) . . . (m + l 2 -m 2 ) 
where n —m is an odd number, we find 
d 2 1 d l — m2 \n n 2n 2 + 2n — 2w 2 — 1 . (n 2 — m 2 )(n — l 2 — m 2 ) 
d %~~ + jhh— ^»+a+ (2n-l)(2n+S) ^ <+ (4h 2 -1)(4?T^1 2 -1) ^ 
It appears therefore that VL' can be expanded in a series of the form 
-- -+ dqQ;« + 5 .... 
MDCCCLXXX. 
s 
(33) 
