128 
PROFESSOR C. NIVEN ON THE CONDUCTION 
Lord Rayleigh has given another form for S„ in terms of differential coefficients 
with regard to x only, but what we are principally concerned with here are the 
expressions (27) which show that S„ alone is finite at the centre, and with equation 
(26) which gives rise to the following formulae of reduction. 
8. Replacing in (26) u n by S„. we find successively 
d, — x 
cP S«_ Q , 2n+l 0 f n(n— 1) 0 
7 o I 1 ' I 0 
CtX X X" 
If we substitute these expressions in the equation which S„ satisfies, we find 
Whence 
and finally, 
S„ 
s„ +i +^s, +1 +s,=o. 
+ + -9, 
X 
(2 n + 1) ~ = (n-\- 1)S, /+1 — «S„_ 1 ; 
S„ + , 
2S„ 
8,;0 
(2n + l)(2?i + 3) (2 ti- 1)(2»+3) 1 (2w-l)(2?H-l) 
1 dS H 
n H1 o ■ . ?iS K _o 
'"+2 + ( 0 n _ 1 VO,, . q\ ■+■ (On — 1V 2m 
> • • (29) 
(fa (2»+1)(2 m+ 3) " +a ' (2n-l)(2n + 3) ' (2»-l)(2n+l) 
We have to substitute these values in 
which may also be written 
d 2 S„ 1 dS„ , c m- \ 
n(n-\- 1) — 1 cl S, t 
o Oyj' ~Z 
x“ x dx 
0 0 
{n + Yf—iid 0 2rP + 2n— 2m 2 — 1 0 n*—m* c 
TR TV 1 \ /0^, i Q\ i \ /r> m 1 
(2» + l)(2» + 3) 
(2n-l)(2n+3) 1 (2w-l)(2»+l) 
If we now write 
S„= 
1.3.5 . . . (27^-1) 
((n — l)~ — m~)(n — 3~—vv l ) . . . (m 
• (30) 
^ ^ 2/r+ 2/;—2/;r — 1 v (»~-m 2 )(»-l~-//r) v /q,\ 
Py~" +3 " t " (2n-l)(2n+3) (4n*-1)(4^=1*-1) 1 ’ 
