246 
ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. [37 
Similarly (a, Bq) — \ { ( 1 + A 2 ) (/am + w) — (Xu + ct) (A/a — v) 
+ (A,/A — V ) (llV + -sr) — ( Xv + w) ( 1 + /A 2 ) 
+ (vA + fi ) (/aw — u) — (— w + Aw) (fiv + X )} 
ie. (a, Bq) = 0, (43), 
and similarly (a, Or) = 0 ; 
whence (a, h ) = 0, and therefore (b, h) = 0, (c, h) = 0 (44); 
also (Jc, h ) = 0, (45). 
Next we have to determine (a, e), (b, e), (c, e). Here e being a function of 
u, v, w, X, /a, v, a, b, c, h, we must write 
(a, e) = {(a, e)} + (a, b) ~ + (a, c) j q + (a, h) ^, 
/ \ (/ m t de de 
ie ' (a, «)={(«, e)) + b Jc —c^. 
But e=i — 2 ; and thence {(a, e)} = —^-(a, v), 
and v is given immediately as a function of X, /a, v, u, v, w, by the equation (38). 
Hence 
(a, u)={[ ( 1 + A 2 ) {(1 + k) utb + Aot 2 } — ( Xu + -sr) [u + A (1 + k) ot} 
+ (A/a — v ) {(1 + k) wit + /aot 2 } — ( Xv + w) [v + /a (1 +• k) ct} 
+ (yX + /a) {(1 + k) wct + V'st' 2 )} — (— v + Xw) {w + v (1 + k) ct}] 
= i {(1 + «) uru — A (1 + k) tx 2 + Xk — X (u 2 + v 1 + w 2 ) — uw] 
= [kuvt — Xrx 2 — A (u 2 + v 2 + w 2 )} 
= Ikuvt-— =i(n M - —) {by (37) and (33)}, 
K \ K> J 
whence 
The terms 
= ^ (bOr — cBq) 
{(a, e)} = -y (bCr-cBq), 
/ \ I.-, sy T1N1 
(a, e) =- v (bCr-cBq)+bj c -c^. 
b^-c^ are evidently of the form F(v) — F(v 0 ). 
(46); 
If therefore we suppose v = v 0 , we have 
(a, e) = - ^ (bGVo - cBq 0 ) 
V o 
(47).