312 
MR. G. T. WALKER ON REPULSION AND ROTATION 
Now if H n be periodic in the time 27 rjp, it can be expanded near to the surface of 
the sphere in the series 
t (-) [Y„ cos pt + Z„ sin pi] 
n= 1 \®/ 
where Y„, Z, t include only harmonics of the 74 th degree. 
Due to this value of H 0 we shall have at the surface 
— - [Y„ cos pt + Z„ siny>t].(x.), 
and 
[ 47r P« ( cos P t + sin pt) 
+ 2n + 1 cr (Y„ sin pt — Z„ cos p£)] . . . (xi.). 
The mean value of sin 3 pt and cos 2 pt being and of sin pt cos pt zero, that of 
3<f> 
[K 
d(j) 
will be, denoting differentiation to <f) by dashes, 
Iff 8 
n 
In + 1 pa 
-X -Y n)t 
\ct ] n + 1A„ 
( irrpaY'n — 2n + 1 crZh) 
In „ \ 2n + 1 pa - 
- S U Z " 2 + 2» + 1 «Y.) 
Now Y' n . z; are harmonics of the ?4 th degree and will give zero when multiplied 
by harmonics of other degrees than n th and integrated over the sphere : hence it is 
sufficient to write for the couple 
4 [f ds i i? [( 2 » + 1 )<T (Y„Z'„ - Y'„Z„) - iirpa (Y„Y'„ + Z„Z',)]. 
Also Y„ (0Y„,/3(/j) being a perfect differential, vanishes when integrated to </> from 0 
to 277 - : thus 
clS . Y„Y'„ = 0, 
ffrfS . Z„Z'„ = 0 , 
and the couple is 
ffrfs 
, 2(n + l)A„ JJ 
' 34, _ 7 3Y,' 
_ " 3</> " 3</> 
• • • • (*£)• 
