322 
MR. G. T. WALKER OK REPULSION AND ROTATION 
On multiplying out and replacing products of sines and cosines by sines and cosines 
of added or subtracted angles most of the terms cancel, and we are left with 
\ sin 2 (y — y) + f sin 2 (a — y — y'). 
Thus the couple on the a, b sphere is 
12 r ir i p 2 a i a' i aa' 
K > p"~\x lC p 
and on the a', b' sphere, 
[sin 2 y — y + 3 sin 2 (a — y — y)\ 
, ‘i2'jr i p 2 a i a' i acr' r . , . t 
K '~pY°-\ AG r[_ sm2y- r + osm2(»-y - y)]. 
25. From this we see that 
(a.) If the couples on the two shells be equal and opposite 
(a — y — y) = 0 or dz ^ tt, 
i.e., 
a = y + y or y + y 7 dz •§- 7r. 
(/3.) The couples will not vanish when c = 0 (and y = y = 0), unless in addition 
a = 0 or ztz 2 tt (in which case there is by symmetry obviously no couple). 
(y.) We may take as an example 
y = 30°, 
7 
= 60 c 
and the couples will be 
727r 2 p 2 rda / Vo- r , s/% 0 . 0 n 
.yxx ,,- [± — - 38111 2 *I 
the upper sign referring to the a, 6 shell. 
The bracket will be positive when a = — 45° say. 
The bracket will be positive and negative when a = 0 or 90°. 
Fig. 43. 
