OP FLUIDS ON THE MOTION OF PENDULUMS. [49] 



which gives 



R ^r-x ^ = 2 cos9 (Ar-* + Br- 1 + C + Dr*), 



r 2 sin ad 



9 = ]—x ~ = sind (Ar~ 3 - Br~ x - 2(7 - 4,Dr 2 ). 



r sin d dr 



The first of the equations of condition (117) requires that 



D = 0, C=-^V. (120) 



It is particularly to be remarked that inasmuch as the two arbitrary constants C, D 

 are determined by the first of the conditions (117), none remain whereby to satisfy the 

 second. Nevertheless it happens that the second of these conditions leads to precisely the same 

 equations (120) as the first. The equations of condition (116) give 



whence 



R= - v { 1 -r r + t?) cose > (,22) 



_, / 3a a 3 \ . 



e = v { l -rr-^) sme < 123 > 



If now we wish to obtain the solution of the problem in its original shape, in which the 

 sphere is in motion and the fluid at rest, except so far as it is disturbed by the sphere, we 

 have merely to add Vco&O, - VsinO, ^ Vr* sin 2 to the expressions for R, 0, \k. We 

 get from (121) 



* = F« 2 (^-J)^e (124) 



40. Let us now return to the problem of Section II. ; let us suppose the time of 

 oscillation to increase indefinitely, and examine what equation (40) becomes in the limit. 



When t becomes infinite, n, and therefore wi, vanishes ; the expression within brackets 

 in (40) takes the form 00 — 09, and its limiting value is easily found by the ordinary methods. 

 We must retain the m 2 in the coefficient of t, because t is susceptible of unlimited increase. 

 We get in the limit 



y], = ± ; a !! ce' i ' m ' t (— --) sin 2 (125) 



dP 

 If now we put V for -~ , the velocity pf the sphere, we get from (39), ce' l ' m ' t = V. After 



substituting in (125), the equation will remain unchanged when we pass from the symbolical 

 to the real values of y\, and V, and thus (125) will be reduced to (124). 



* I have already had occasion, in treating of another sub- I I had obtained as a limiting case of the problem of a ball 

 ject, to publish the solution expressed by this equation, which | pendulum. See Philosophical Magazine for May 1848, p. 343. 



Vol. IX. Part II. 31 



