145 
404] 
ON THE ROTATION OF A SOLID BODY. 
Hence the expression for sin j sin </> dò is 
— — w -2 s i n j [{P cos </> - sin cf) (r cosm - q cos rc)} dp+ ...] 
dco r , 
+ o,*sinj C0S ^ ~ sin ^ cos m ~ 2 cos n )} P + • • •] 
o)^sin j ^ C0S ^ “ s ^ n $ K r cos m — <1 cos n) dp + ...}] 
H r—r~ — ft) 2 COS <f> 
ft) * Sin J 
= sinj sin <p d(f>, 
or finally 
• • • j 7, 1 sin<f> r/ s , „ „ 
sm j sm 9 d(p = — 7 —r [(r cos m — r/ cos ?i) cip + &c.] 
that is 
1 ( (r cos m — q cos n) dp ) 
w sin j , _|_ (p cos n _ r cos i ) dq y 
( + (q cos l —p cos m) dr j 
which is the required expression for d<j). 
Recapitulating, A, B, C, p, q, r denote as usual, 
r B-C nr C-A , r A-B , 
L = —- — , M = —-g—, N = —^— , du = pqrdt, 
p = V (21 + 2 Lu ), 
g = V (23 + 2Mu), 
r = V (6 + 2ÌV+) ; 
21 + 23 + 6 =#, 
2{A + 23R +6(7 
2U 2 + 23R 2 + 6C' 2 = (7; 
Z/^236 + ili 2 M2( + ÌW2123 =H, 
DA*m + i¥ 2 R-’62l + iY‘-’(7-2l23 = if ; 
so that 
K = EG - F 2 , 
(.H-2LMNFu) du*J{G) 
(K - 2LMNGu) V {(21 + 2Lu) (23 + 2Jfw) (6 + 2Nu)) ’