6] ON THE MOTION OF ROTATION OF A SOLID BODY, 
to which are to be joined 
tcp = 2 ( 
33 
dX 
dp 
dv\ 
dt 
+v dt 
St) 
dX 
dp 
_ dv\ 
dt 
+ St 
+x dt) 
dX 
dp 
dv\ 
dt 
~ X dt 
+ SiJ 
where it will be recollected 
k — 1 + X 2 + p 2 + ir ' 
and on the integration of these six equations depends the complete determination of 
the motion. 
If we neglect the terms depending on V, the first three equations may be 
integrated in the form 
s 2 G-B „ . A — C 
P = Pi ~ - j— </>’ ( f = fr 2 r- </>, 
2t = f 
B 
dcf) 
r a + n 2 - 
B-A 
G 
0, 
G-B 
Pi 
<f> ( 1\~ 
A-G 
B 
0 J *i* ~ 
B-A 
G 
and considering p, q, r as functions of cf>, given by these equations, the three latter 
ones take the form 
k _ dX dp dv 
4qr dcf) JrV dcf) ^ dcf)’ 
k _ d\ dp dv 
4rp V dcf) + dcf) + d(f> ’ 
k dX dp dv 
4pq p dcf) dcf)+ dcf>' 
of which, as is well known, the equations following, equivalent to two independent 
equations, are integrals, 
fcg = Ap (1 -f A 5 — p- — v~) 4- 2Bq (Xp — v) + 2 Gr {vX 4- p), 
Kg' = 2Ap (\p 4- r) + Bq (1 + p 2 — X 2 — v 2 ) 4- 2Gr (pv — A), 
Kg — 2Ap(yX — p) -\-2Bq(pv + X) + Gr (1 + v 2 — X 2 - p 2 )\ 
where g, g, g", are arbitrary constants satisfying 
g-+g- + g"- = A-p;~ + B-q 2 + GVj*. 
To obtain another integral, it is apparently necessary, as in the ordinary theory, to 
revert to the consideration of the invariable plane. Suppose g' = 0, g" = 0, 
then g" = d (A 2 p x 2 + B‘-q 2 + C 2 i\ 2 ), = k suppose. 
C. 
5