328 
MR. GL T. WALKER ON REPULSION AND ROTATION 
Expressed in terms of a, the couple on the two shells will be 
2 y — 2y'. 
29. Hence we have 
(a.) If y — tt — y, 
(f) (a, y, y) = 3 sin 4y + 2 sin 2y cos 2a — 8 sin 2a. 
For the couples to be equal and opposite we must have a = 0 or t, and then 
<f> (a, y, y) = 3 sin 4y dr 2 sin 2y. 
-K 
„ 2 ? 7 r'rPcfta'^ocr' . . 
- K 2 
277r+ 3 a%' 4 crcr / 
<M a > y'» y)> 
where 
(f> (a, y, y') = sin 2a + 3 (sin 2y + sin 2y') + sin 2a — 2y' 
— sin 2a — 2y — 3 sin 2y' — 2y — 9 sin 2a — 
If y — ^ u (fig. 54), the values of (a, y, y) are negative, both for a = 0 and 
a = ^ tt ; if y = ^ 77 (fig. 55), both values of <£ (a, y, y) are positive. 
(j3.) If we take y = 30°, y — 60°, we find that 
/( a > 7> V) 
f( a > y, y) 
’ 3\/3 
2 
'V3 
+ 9 sin 2a 
+ 11 sin 2a • 
The former is negative when a increases from 0° to about 98° 23', and positive 
thence to 171° 37', being negative to 180°. The latter is negative from 0° to 112° 34', 
positive thence to 157° 26', and afterwai’ds negative to 180°. 
