318 
MR. G. T. WALKER OK REPULSION AND ROTATION 
D' = — l ^r I + terms three degrees higher, 
tt ./ _ 12irpa n a'b ' 
n^A' -L . • . 
P A 1 
Hence the couple increasing 0 is 
67 Tfd 
or 
~a 2I«V 2 127rpa'~a' T , 7 , 7 x 
a, • 1 (c * + 26c) 
1447r 3 w 2 cra- / a%'' 4 C , 7 , ,,, TO 
^wT (26 + 6)i - 
22. In obtaining the couple on the a, V shell it will not do merely to interchange 
dashed and undashed letters, for the equations (xiii.) give when r = ci and < cl, 
,-1 — t 
so that 
1 -f- sin O' cos (f)' 
sin 0 cos 0 1 
~ = d* 
r~ 
1 -f- j sin O' cos 0' . . . 
[ — cl -j- ci sin O' cos </>'] 
1 2a ' • a x’ 
=-^ — 77 sm 0 cos 0 , 
d- d 6 
sin 6 sin 0 a' . A/ . , 
— - = sm 0 sm 0 , 
cos 0 a 
—T-- = — cos 0. 
r- d;‘ 
These equations may be obtained from (xiii.) by interchanging dashed and undashed 
letters, leaving the sign of d unaltered, with the exception of the term — (1 /d 2 ) in 
(sin 0 cos 0)/r 3 , and as this term does not appear afterwards (being constant over the 
sphere), the exception is negligible. 
The subsequent work does not introduce d afresh, it only makes use of the formulae 
we have obtained, and thus it will be seen that the final couple on the a, b' shell is got 
by the changing of dashed and undashed letters, leaving the sign of d unaltered ; it is 
1447T 2 p 2 crcr , ft 4 ft' 4 C 
P y 2 <P \ A\ 
(b + 2b’) I 2 . 
If we write h for 
couples are 
\ (b -f- b'), the mean of distances of the centres from 0, the 
7 27 T 2 ^ 2 crcr'ft 4 ft ,4 c 
pf'fp \ A\ 
(6h ± cl) I 2 . 
