250 ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 
Again, ( h, 8) = 2p {Ap, 8) + 2q (Bq, 8) + 2r {Gr, 8), 
{Ap, 8) = {Ap, 8') + {Ap, 8"), 
(Ap,B') = ^(Ap, «v) + (Ap, 
= — (Ap, tew), 
KV 
{Ap, /cut) = \ { ( 1 + X 2 ) {kU + 2Xht) — {Xu + ot) kX 
+ {Xfi + v) {kv + 2/u,'scr) — {Xv — tv) k^ 
+ (vX + [JB ) (KW + 2y-57 ) — { v + Xtv) kv } 
= ^ k (u + Act) = ^ k (a + Ap) (55); 
therefore (Ap, S’) = ^(a, + Ap) (56), 
(Ap, S") = -k h ^~-(Ap, v) + &e. 
P h ^~- (bGr - cBq) + Fv- Fv„ 
(Ap, B) = L ( a _ Ap — h^^ (hCr — cBq)} + Fv - Fv 
and similarly for {Bq, 8), {Gr, 8). Substituting, and neglecting the terms which vanish 
for v = v 0 , 
(h, 8) + v) , 
i.e. (h, 8) = 0 (57). 
Lastly, to find (e, 8), 
<«. 8) = {(<r, 8)) +(a, 8)^ + (b, 8)^ + (c, 8)^, 
where, in {(e, 8)}, the differentiations upon e are supposed not to affect the constants 
a, b, c. Neglecting the terms which vanish for v = v 0 , 
{e, 8) ={(e, 8)}, 
{(e, 8)} = {(e, 801 +{(e, 8")}, 
{(«, «0} = [{(«. 8')}] + fe «'))]; 
where, in [{(e, S')}], the differentiations upon e and 8 do not affect the constants. 
{(«, «")}-[{(«, «")}] +(*> a )^ + &c - 
{(«, «")}-[{(«. «"))]• 
i.e.