6] ON THE MOTION OF ROTATION OF A SOLID BODY. 31 
where X', /.xv, denote Reducing, we have 
rep = 2 (A/ + vp — v p): 
from which it is easy to derive the system 
Kp = 2 ( X' + vp — v'p), 
re(£ = 2 ( — i/X + p v X), 
xr = 2 ( /xX' — X/x' + v ) ; 
or, determining X', p, v, from these equations, the equivalent system 
2X' = (1 + X 2 ) p + (X/x— v ) q + (i/X+ p) r, 
2p' = (X/x + + ( 1 + /x 2 )g + (pv— X) r, 
2if = (i/X — p)p + (pv + X)q + (l +v 2 )r. 
The following equation also is immediately obtained, 
k — re (Xp + pq + vr). 
The subsequent part of the problem requires the knowledge of the differential 
coefficients of p, q, r, w 
down the six 
respect 
to X, p, 
. -\ t / / 
v, X, p, v. 
It will 
II 
2, 
o, 
dq 
K dX'~' 
-2i/, 
K È. + 2 ^ = 
2i/', 
dr 
K dX~~ 
2* 
dr 
re -j- + 2rX = 
dX 
- 2/x', 
from which the others are immediately obtained. 
Suppose now a solid body acted on by any forces, and revolving round a fixed 
point. The equations of motion are 
d dT_dT = dV 
dt d\' dX dX ’ 
d dT _dT = dV 
dt dp dp dp ’ 
d dT_dT^dV. 
dt dv dv dv ’ 
where 
T= \ (Ap- + Bq 2 + Gr 2 ) ; V = 2 \J{Xdx + Ydy + Zdz)\ dm ; 
dV dV dV 
or if Xdx + Ydy + Zdz is not an exact differential, ^ ^ 
symbols standing for 
, are independent 
*( x £ +Y ï+ z £) dm ’