ON THE ROTATION OF A SOLTD BODY ROUND A FIXED POINT. 
239 
37] 
where 
p -i 
Q 
1 
B 
«=e 
(B-G)qr + i {( 1 + V>^ + (V* + »)g+(to-,)*} 
{ 0-A)rp + i\(j,X-v) ^+(1 +^)^ + (^ + >)^} 
(A- B)pq + i j(,X + f) (/»> - X) ( 1 +,») 
4 
(2). 
A = i K 1 + ^- 2 ) V + (ty- — v)q + (\v + p) r}, ] 
M = i {0^ + v) p + ( 1 4- p 2 ) q + (pv — \)r} } l (3). 
N = l {(i/\ — p) p + (pv + A) q + ( 1 + v 2 ) r],J 
[whence also AA + ¿¿M + = £ k (\p + pq + vr) (3 bis)]. 
In the case where the forces vanish, the first three equations become simply 
P = ^(B-G)qr, j 
Q=±(C-A)rp, 
R = Q (A-B) pq, 
(4), 
and here the usual four integrals of the system are 
Ap 2 + Bq 2 + Cr 2 — li (5), 
Ap (1 + A 2 — p 2 — v 2 ) + 2Bq (Ap — v) + 2Or ( v\ + p) = a (L + A 2 + pi 1 + v 2 ), 4 
2Ap (Ap + v) + Bq (1 + p? - v 2 - A 2 ) + 2Cr (pv - A) = b (1 + A 2 + p 2 + v 2 ), ; (6), 
2Ap (vX — p) + 2Bq (pv + A) + Cr (1 + v 2 — A 2 — p 2 ) = c (1 + A 2 + p 2 + v 2 ), J 
or as they may also be written, 
a (1 + A 2 — p 2 — v 2 ) + 2b (Ap + v) + 2c (vX — p) = Ap (1 + A 2 + p 2 + v 2 ), 4 
2a (Ap — v) + b (1 + p 2 — v 2 — A 2 ) + 2c (pv + A) = Bq (1 + A 2 + p 2 + v 2 ), y 
2a (vX 4~ p) -(- 2b (pv — A) 4- c (1 v 2 — A - — p~) = Gv (1 4" A - 4- p~ 4~ v“'), J 
(6 bis); 
to Avhich we may add, 
A 2 p 2 4- B 2 q 2 + G 2 r 2 = k 2 (7) ; 
where k 2 = a 2 4- b 3 4- c 2 (8). 
Introducing the quantities k, O, (the former of which has been already made use 
of) given by the equations 
x = 1 4- A 2 + p 2 4- v 2 , ] 
G • 
XI = A Ap + pBq 4- vCr, J