324 
PROFESSOR C. NIVEN ON THE INDUCTION OF ELECTRIC 
a being the mean radius of the shell (sensibly constant), k a finite quantity : 
the passage clear we know that, if s —sin 9, P,/ satisfies the equation 
„ 9 ,\ (p y , t- 2 ^ 2 dy 
w+— -.+ 
ds 
!)■ 
77 V 
'7 
y —0 
(see Heine, ‘ Kugelfunctionen,’ p. 216, 2nd ed.) ; which we may satisfy by 
y = C ; /(s'"+A 1 S“ +3 + A s S* +4 + . . . ) 
where 
A 1 {(w+2) 2 —m 2 } +n(n +1) — to. (to+ 1) = 0 
A 3 {(w+4) 2 —m 2 } +[n(n-f- 1) — (to+2)(to+3)]A : =0 
A 3 { (to+6) 3 —to 2 } 1) — (to+ 4)(to+ 5)] A 2 = 0, &c. 
If we now put n (infinitely great) 
= «:a, s= 
we find 
cl 
C w 
P » 
a™ r 
o o 
K 'P 
K i p i 
K (i p 6 
4. m +1 4.8.to + 1.to + 2 4.8.12.to+1.to + 2.to + 3 
2 “m! 
K m 2j n 
C m n .J m (Kp) 
the value of the constant C,/ being obtained from considering the value 
when 5=0 ; in fact we have (Heine, 2nd ed., p. 207), 
CL"=2 
n\(n + m )! 
i—m \ ' 
m ! (2n )! 
This constant we shall keep, for the present, unreduced. 
From the theorem PA-PA' sin 9d9= 0, we derive at once 
Jo 
I J m (Kp)J m (Kp)pdp—0 
- 1 0 
We have moreover (see Heine, pp. 327 and 253), 
f (PA) 2 sin 6d6 
J n 
2 (n + to)! (n — m )! 
2w + l*(1.3.5... 27 -Tv 2 
and, correspondingly, 
pyy— 7 /c 2 '" l a 2m+2 2 (% + to) ! (%—to) 1 
Jo P 1 P— (C m ») s "2w+ l’(1.3.5 ... 27i-l) 2 ‘2 3 »(TO !) 3 
to make 
• (A,) 
• (A 3 ) 
P n • 
TO • 
• (A*) 
• (A 6 ) 
• (A 6 ) 
