ON THE MOTION OF PENDULUMS. 29
lent V^"(r), and then integrating. It is unnecessary to add an arbitrary function of the time, since any such function may be supposed to be included in IT. We get
wcos6'/1/(r) ..................(37).
16. The integration of the differential equation (33) does not present the least difficulty, and (34) comes under a well-known integrable form. The integrals of these equations are
1
!rx .ixl- ............(38)>
^ = (7e~W V -mr/- V mr,
and we have to determine J., B, 0, D by the equations of condition.
The solution of the problem, in the case in which the fluid is confined by a spherical envelope, will of course contain as a particular case that in which the fluid is unlimited, to obtain the results belonging to which it will be sufficient to put 6 = GO . As, however, the case of an unlimited fluid is at the same time simpler and more interesting than the general case, it will be proper to consider it separately.
Let + m denote that square root of fju'~l n J — 1 which has its real part positive; then in equations (38) we must have D = 0, since otherwise the velocity would be infinite at an infinite distance. We must also have .5 = 0, since otherwise the velocity would be finite when r = co, as appears from (26). We get then from the equations of condition (35)
A = ia?c + ^(l + -1) 0 = -— (T 2iTfi \ ftnaj 2/fn
whence
(/ ^ *} \ n
• 2/31/1 i " i O \ to
sin cN H-------1—2"o -
(V ma . maJ r
- — fl + — )e-"'(r-0'^ ..................(40),
ma \ mr/ '
3
p = ipoc/*'m« fl + — +4^) e-^'cos 6 ^ r r \ ma ma J r