78] NOTE ON THE MOTION OF ROTATION OF A SOLID OF REVOLUTION. 
463 
From the equations (12) and (15) in the second of the papers quoted, we deduce 
2v = k [k + A (n — v) sin] + MA cos ] cos (6 + ¿)}, 
<3> = k {n sin] + M cos] cos (6 + ¿)}, 
V = — vkMA cos] sin (6 + i), 
(values which verify as they should do the equation (19)). Hence, from the equation (27), 
writing = dt = ~ d6, we have 
v 
h + kn sin] + kMcos] cos {6 + i) 
k + A (n — v) sin] + MA cos] cos (6 + i) ’ 
This is easily integrated; but the only case which appears likely to give a simple 
result is when the quantity under the integral sign is constant, or 
A (h + kn sin ]) = k [k + A (n — v) sin]}, 
or 
Ah — k 2 + Akv sin] = 0 ; 
that is, 
A (n — v) + k sin] = 0 : 
whence 
Observing that \ir — j is the inclination of the axis of z to the normal to the invariable 
plane, this equation shows that the supposition above is not any restriction upon the 
generality of the motion, but amounts only to supposing that the axis of z (which is 
a line fixed in space) is taken upon the surface of a certain right cone having for 
its axis the perpendicular to the invariable plane. Resuming the solution of the problem, 
we have 
which may also be written under the form 
where