528 
PROFESSOR H. LAMB ON ELECTRICAL 
Outside : 
a— 
b= 
( i -i \dX,j dX_ H , 
K+1) *r +B ^r 
/ i i dX_ M _^ 
(35). 
In deducing (34) we have made use of (29), (30), and of the known formula 
xy — r3 (<h&_ r %n+\!L r 
X>1 2w+1\ dx dx Xn 
2«-l 
. . . (36). 
We have now to apply the conditions to be satisfied at the surface of the sphere. 
If It be the radius, the continuity of F, G, H requires 
(37). 
The continuity of a, b, c requires 
^_!(^R).X« = X„.(38)," 
with another condition which is, however, implied in (37) and (38). We must bear 
in mind that in these equations r is supposed put = R throughout; so that X,„ 
X_„_ 1 are now surface harmonics, of order n. 
Second type. We have 
Inside the sphere : 
.(39) 
F= 
G= 
H= 
1 d(f>„ , / , , x , n fo) u 7j2 r 2»+3 ( /7 x d 
~A ^+( w + 1 )^*-lW/7^"" n 9»-i-i V cr )jJ a * r " 
clx 
At + 1.2?i + 3 
/,2 r 2;;+3 
'271 + 1.271 + 3 
-2«-l 
1 d(f) H . . . \ r / 7 
1 , /7 /a r 2«+3 <7 
dx 
2?i + 1.2;i + 3 ‘ 
■ (40).t 
* The vigorous proof of these, ancl of similar inferences in the sequel, may he conducted as in § 4 of 
the paper “On the Vibrations of an Elastic Sphere” already cited, 
t These may also be written 
F= ~ldt + (2^ + 1 )^- 1 (^)^7-’4 { *«(*»■>».}. 
