6] 
ON THE MOTION OF ROTATION OF A SOLID BODY. 
35 
Note. It may be as well to verify independently the analytical conclusion imme 
diately deducible from the preceding formulæ, viz. if X, ¡x, v, be given by the differential 
equations, 
where k = 1 + X 2 + ¡x- + v 2 , 
particular values of X, [x, 
given by the system 
dX 
+ v 
d/x 
dv 
Kj) = 
dt 
dt 
-Pdt’ 
dX 
d/x 
^ dv 
Kq — — 
V dt 
+ 
dt 
+ x di’ 
dX 
dfx 
dv 
KV = 
^dt 
— \ 
dt 
+ dt’ 
q, r, 
are 
any 
functions 
be 
v, and l, m, n, arbitrary constants, the general integrals are 
Po =1 — lX 0 — m/x 0 — nv 0 , 
P 0 \ — l + X 0 + niv 0 — 7ifX 0 , 
P 0 fi = m + fx 0 + n\ 0 — Ivo , 
P 0 v= n -f v 0 4- l/xo — mX 0 . 
Assuming these equations, we deduce the equivalent system, 
(1 + XX 0 + fifi 0 + vv 0 ) l = X — X 0 + v 0 fx — v/x 0) 
(1 + XX 0 -f fx/x 0 + vv 0 ) m = fx — [xo + X 0 v — Xv 0 , 
(1 + XX 0 + /x/x 0 + vv 0 ) n = v — v 0 + /x 0 X — /xX 0 . 
Differentiate the first of these and eliminate l, the result takes the form 0 = 
(/V “t Vo) X “b vfx v fx) (v 0 X 0 fx 0 ) ( vX + [x + Xv) + (/x 0 + X 0 v 0 ) (jxX — X/x + v ) + k 0 X , 
+ {N + V-) (Xo + VofXo — V 0 'fXo) + (v — X/x) ( — V 0 \' 0 + fXo' + X 0 V 0 ') — (fX + Xv) (jxX — + Vo) — kXo, 
where X', &c. 
denote ~ 
at 
&c. and Kq — 1 -f- Xo” -b [Xçf ~b Vq~. 
Reducing by the differential equations in X, /x, v\ X 0 , /x 0 , v 0 , this becomes 
k 0 [X' + (/x 2 -b v" ) -b \q (y - X fx ) - (jx + Xv )} 
— k {X 0 + \p (/v + v 0 ~) -f \q (y 0 — X 0 /x 0 ) — |r (fx 0 + Xc 0 )| = 0 ; 
or substituting for X', Xo, we have the identical equation 
(/c 0 «: — kk 0 ) = 0 : 
and similarly may the remaining equations be verified.