37] ON THE ROTATION OF A SOLID BODY ROUND A FIXED POINT. 
245 
standing for Xu + pv + vw. {These determine №, which is however given immediately by 
(37).} As for h, we have 
where Ap, Bq, Cr are given as functions of A, p, v, u, v, w by (32), in which also vr 
stands for Xu + pv + vw. Again, 
in each of which V, 4> are functions of v, and of a, b, c, h, partly as entering explicitly 
into these functions, partly as contained implicitly in p, q, r, which enter into V, cp, and 
are functions of v, h, k given by (18). After the integration v is to be considered a 
function of A, p, v, u, v, w given by (38). Both of the integrals may be supposed taken 
from a certain value v 0 of v, which may be considered as an absolutely invariable arbi 
trary constant, since without it we have the right number, six, of arbitrary constants. 
First to find (a, b), (b, c), and (c, a). From (34) we have 
(a, b) = J { ( 1 + A 2 ) (pu — w) — (Au + ■&>) (Xp + v) 
4- (Xp — v ) (pv + vr) — (Xv + w) ( 1 + p?) 
+ (vX + p) ( u + pw) — (Aw — v) (pv — A)} 
= ^ (pu — Xv — w — vct) = — ! 2c = — c ; 
whence the system 
(b, c) = — a, (c, a) = — b, (a, b) = — c 
(40). 
Also we may add (k, a) = | (a, a) + ^ (b, a) +1 (c, a) = 0, 
or 
(k, a) = 0, (Jc, b) = 0, (k, c) = 0 
(41), 
which will be useful in calculating some of the following coefficients. 
Proceeding to calculate (a, h), (b, h), (c, h). It is seen immediately that 
(a, h) = 2 [p (a, Ap) + q (a, Bq) + r (a, Cr)}, 
where Ap, Bq, Cr, are given by the equations (32), so that 
(a, Ap) — { ( 1 + A") (Xu 4" ot) — (1 + A“) (Xu -t- ot) 
4" (Ap — v ) (Xv — w) — (Ap -f v) (Xv -H w) 
4“ (vX 4~ p) ( v 4" Aw) — (vX — p) (— v 4" Aw)} 
i.e. 
(a, Ap) = 0. 
(42).