OF HEAT IN ELLIPSOIDS OF REVOLUTION. 
127 
or 
dHiOrx) , 
~ -\- UqX —0, 
. sm x , cos x 
u 0 =A— +B—, 
The two values of R are therefore those given by (25). 
We may readily find these values as expansions in powers of x, by putting t for x 2 
-i v sin Jt . cos \/t 
and expandmg — Jf. - and ^ . 
q _ (-1)’^” 
' ““1.3... (2w4 
1 — 
(2to41)L 1.(2w43) 2 1 1.2.(2a + 3)(2?i + 5) 4 ' ' 
_(—1)“1.3 ... (2n— 1) n ,1a; 2 . 1 
,„+i i 1 “r 
1 
l.(2n — l) 2 1.2.(2w-l)(2w-3) 4 ‘ * J J 
r* • (27) 
Since the differential equation is unaltered by replacing n by — (n + 1), it follows 
that S_„_ 1 and T_, ; _ J are also solutions of it; and in fact it is clear, in comparing 
corresponding terms, that 
S_„_ 1 =( —1)“T„ 
T_„_ 1 =( — 1)"S„, 
the constants, introduced by integration in S_„_ 1} being so adjusted as to make the 
first n + 1 terms agree ; no constants are to be introduced in determining 
—/ cl \ n s 
The form S // = 2'T 2 (—j • - 
quoted by Lord Rayleigh, 
— / (l \ ti sin a/ ' t • » . 
The form S, / = 2'T 2 ( — j • ^ also indicates the expansion as a Bessel’s function, 
so 
S«=(—1)“ \/\ x ; 
T„=( —1)“ /y/|aj- 4 K„ +i (aj). 
The finite expansions of S„ and T„ are 
g _/l n\n'-V) 1 , n\n'-V)(n'-2')(n'-S’) J. 
+ 
1.2 2h? 1 
n' 1 n\n'-l'){n'-2') 1 
1.2.3.4 
04,,.5 
■I I H7r 
sm (^ 
1 2x- 
1.24 
W4+ 
nir 
COS I 2C + —- 
>• (28) 
T (1 n'(n' — 10 1 
1,1 \x 1.2 ' 24k 3 ' • 
m t 
cos [x+ — 
n' 1 n'(n'— V)(n' — 2') ] , \ • i i nir \ 
7 '2tf TJ .3 
2 -v+ • • • ) s in (*+y j 
in which n stands for n(n-\- 1), and n — r= (n — r)(n-\-r-\- 1). 
