OF HEAT IN ELLIPSOIDS OF REVOLUTION. 
125 
C(« 0 P^-« 1 P i /' + ' + r/,P ? / +4 - . . . ±a i P m M -\- . . . ), 
or by the expression 
V(b ( p m m+1 -b 1 P m m+3 + . . . 
in which 
I. 
y • • • ( 2i ) 
±aJ? m m+ *' +1 + 
V\ a \ — ~{ K Q~fy a () 
II. 
P 1^1— A K 0 % 
— (/q k'jcty O/q P i ^ )^i 6o 
V?Ph— £ ( K -2~k) CL 2~ a l 
Pr+\®r+\ — (/->v ' L—1 |* »+l^+l — ^ ( K j 1 
wherein \ 2 c 2 = e, 
2% 2 +2%—2m 2 —1 , / , , x (% 2 — mF){n— l 2 — m 2 ) , „ 
K<-=—-—--rr-e+n(w + i), p,= n . 2 ^ q , w=m+2r 
(4?i 2 — l)(4m — l 2 — 1) 
P) 
■iy 
( 22 ) 
(2?i —l)(2n + 3) 
/ 2n 2 + 2n—2m- — l , . , (A 2 — m 2 ) (?i — 1 2 —m 2 ) 
Ks ~ (2w-l)(2w + 3) e + ri ( w + 1 )’ ^-(4 % 2 -l)(4.i^I 2 -lY n ~ m + 25 + 1 * 
The arbitrary constants C and D are introduced that there may be no loss of 
generality when we put one of the series a 0 , ci 1 . . . equal to unity, or one of the series 
b () , h 1 . . . . The two sets of formulae I. and II. above are precisely identical, making 
allowance for the difference in the values to be assigned for n, and therefore the con¬ 
clusions drawn from I. will be in general true also for II. ; and on this account we 
shall confine ourselves more especially to the former of these two systems. 
We must first examine somewhat more closely the sense in which the series (21, a) 
satisfies (18). If we stop at the term Ca,P,/ t+2r in forming the expression 
' >t!v‘ iv 
1 0 
— V 
0 +k 2 cV —k p, 
it is not precisely zero, but equal to C( — ea,P w+2>-+2 -|-ea,. +1 P' M+2i '); hence it is only when 
a r and a r+l are either zero or indefinitely small that the differential equation is 
satisfied. 
We have, therefore, to show that, for certain values of k which are definite in 
position, a r tends to zero as r becomes infinitely great. In other words, the values of 
k are the roots of the equation 
(23) 
