37] ON THE ROTATION OE A SOLID BODY ROUND A FIXED POINT. 
241 
Also fi, v are given by the equations (14) as functions of p, q, r, XI, i.e. of v, XI; 
so that every thing is prepared for the investigation of the differential equation between 
v, ft. To find this we have immediately 
civ = | (Ascdp 4- Bbdq + Ccdr) = \ Vdt (19), 
from the equations (4) and (15). V is of course to be considered as a given function 
of v. Again, 
ftdft = \ (rcdv + vd/c) (20), 
where d/c = 2(\d\ + pdp + vdv) (21); 
or from the equations (1), (3), [and (3 bis)\, 
d/c — k (\p + fjuq + vr)dt (22). 
Hence, from (16), 
2vdic — k, {ft (h + d>) — V} dt (23); 
or 2 (vd/c + ndv) = «ft (Ji + d>) dt (24), 
whence 
dft = |- k (h + d^) dt, 
1 ^ + № /7 7, 
— 4 (h -(- d^) dt 
(25), 
and therefore, from (19), 
2did _ h + d> 
ft 2 + k 2 vV 
(26), 
the required differential equation, in which d>, V are given functions of v, i.e. they are 
functions of p, q, r by the equations (15), and these quantities are functions of v by 
(18). The variables in (26) are therefore separated, and we have the integral equation 
2 tan -1 ^ = 8 + k 
(h + d>) dv 
vV 
(27), 
where 8 is the constant of integration. The equation (19) gives also 
t — e = 
(28); 
and thus the solution of the problem is completely effected. The integrals may be taken 
from any particular value v 0 of v. The variable ft may be exhibited as the integral of 
an explicit algebraical function, by recurring to the variable <£ of the paper quoted. 
Thus if 
Ap 0 2 + Bq* + Cr 0 2 = h, 
A 2 p 0 2 + B 2 q 0 2 + CV 0 2 = k 2 , 
Ap 0 a + Bq 0 b + Grp = 2v 0 — k 2 ; 
c. 31