ON THE MOTION OF PENDULUMS.                         33
term for dp/d0 the expression got from (29), and putting r — a, we get
\ V cos 9 sin 6 dd = - *p -y- f* f -.'<T                        ^dt]o\dr
Substituting in the expression for F, we get
sn
19. The above expression for F, being derived from the general equations (27), (28), combined with the equations of condition (30), holds good, not merely when the fluid is confined by a spherical envelope, but whenever the motion is symmetrical about an axis, and that, whether the motion of the sphere be or be not expressed by a single circular function of the time. It might be employed, for instance, in the case of a sphere oscillating in a direction perpendicular to a fixed rigid plane.
When the fluid is either unconfined, or confined by a spherical envelope concentric with the sphere in its position of equilibrium, the functions ^ , fa consist, as we have seen, of sin^2 6 multiplied by two factors independent of #. If we continue to employ the symbolical expressions, which will be more convenient to work with than the real expressions which might be derived from, them, we shall have
for these factors respectively.    Substituting in (49), and performing the integration with respect to 0, we get
(50).
20. Consider for the present only the case in which the fluid is unlimited. The arbitrary constants which appear in equations (38) were determined for this case in Art. 16. Substituting in (50) we get
F = - | irpcfon, J~ ^3   ^r
ma    ma
Putting for m its value v (1+^-1), and denoting by M^r the s. in,                                                                   3