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