244 ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 
these become 2Ap = Aw + u+ vv — ¡xw, (32). 
2 Bq = /¿ct — vu + v + Aw, 
2Cr = ij-ut + ¡xu — \v + w, 
whence also 
2X2 = ktz (33). 
Substituting the values of Ap, Bq, Gr, given by (30) in the equations (6), we deduce 
2a = Act -f и — vv + fxw (34), 
2b = /¿ct + vu + v — A w, 
2c = уст — ¡xu + \v + w, 
whence also 
2 (aA + Ь/х + су) = /ест (35), 
which in fact follows from (33) and (17). And likewise the inverse system, 
2 
и = - ( a + yb — ¡Xc) (36). 
2 
v = - (— ya + b + Ac), 
к 
2 
w = - ( ¡x<x — Ab I - c). 
It is easy to deduce 
A: 2 = ^/с [w 2 + v 2 + w 2 + ct 2 ] (37), 
u = ^ [(m 2 + v 2 + w 2 ) + (1 + к) ct 2 ] (38). 
Again, from the equations (10 bis), 
к (bGv — cBq) = — 2A (a 2 + b 2 + с 2 ) + 2a (Aa + /xh + ус) + 2 (by — с/х) XI 
= — 2A& 2 + 2 (a + by — c¡x) XI 
= — 2A& 2 + icufl; 
and, forming also the similar expressions for к (c Ap — a Gr), and к (a Bq — b Ap), we thus 
obtain 
Пи — - & 2 A = bGr — cBq (39), 
2 
Xly — №/х = с Ap — bCr, 
к 
2 
Пги — ~k 2 v = a Bq — c Ap ; 
to which many others might probably be joined. 
The constants of the problem are a, b, c, h, e, 3. Of these a, b, c are given as 
functions of А, /X, v, u, v, w, by the equations (34); in which ct is to be considered as