336 PROFESSOR C. NIVEN ON THE INDUCTION OF ELECTRIC 
But if X' be also one of the roots of equation (62), both H„ +1 and H ' n+1 vanish at 
the lower limit; and at the upper limit 
2??/ “I - 1 
H„=H , »=l and XH„ +1 =X , H* + i=— K 
hence 
.(69) 
- a 
If we put X' = X+SX, and then make SX infinitely small, we arrive at the value of 
fV 2 HMr : for 
d H„ 
R'*= H,(Xr + §X .r)= H,+r8X ~ 
and 
— H w +-SX(XrH« +1 +?iH„) by (B 4 ) 
8 \, 
H'„ +1 —H„ +1 (Xr+SX.r) — H, i+1 +^(—XrPL— r+ 1H„ +1 ) by (B s ) 
Putting in these values in (68) 
fW ,?dr=: 
{Xr(H„~+H 3 „ +1 )+2nH„EL +1 } 
F(2»+l)-^(S„(Xa)T„_ 1 (Xb)-T„(Xa)S,. 1 (Xb)) i! . (70) 
Let us call this integral l u ; we may now separate the coefficient D' due to any 
term of U„, say cos : for if we put in equation (58) t — 0, P=—P 0 , multiply 
across by cos m<f>dcf>. P,/ sin Odd.R n r~dr and integrate throughout the body of the shell, 
all the terms on the right hand side disappear except those which have the some pi, 
n, and X; and, if we note that 
.1 r !r /f -v 7/1 2 (n+ m) ! (n — m )! T 
C os~ »,«=*•, j u (P„”)- S m • (La57T2^!)5= J - 
we 
find 
D'= — | cos m^dcf) I Vj l . sin 6d0 j P 0 H ;i r‘ 2 ci;“-p7rJ,/I„. 
The coefficient due to sin m<f> P,/ may be similarly found. 
