[28] PROFESSOR STOKES, ON THE EFFECT OF THE INTERNAL FRICTION 

 The equations of condition (30), (3l) become, on putting /(r) for/j(r) + / 2 (r), 



/» = «c, f(a) = ±a 2 c, (35) 



/(6)=0, /(6)=0 (36) 



We may obtain p from (29) by putting for ^ its value e /i ' m ' t sin 2 0/, (r), replacing 

 after differentiation %f\(r) by its equivalent f*f"(r) t 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 II. We get 



p = - pfj.'m i 6 i l ' mH cosef 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 



f,(r) = - + Br\ 



T 



Mr) = Ce- mr (l+—) + De mr (l - —), 

 \ mrj \ mrj 



(38) 



and we have to determine A, B, C, 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 b = oo . 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 (i' J «\/-l 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 B = 0, since otherwise the velocity would be finite when 

 r = oo , as appears from (26). We get then from the equations of condition (35) 



, , 3a 2 e / 1\ _ 3ac 



A=±a 3 c + 1+ , C=-~ 



■ 2m \ ma) 2m 



whence 



£■•,/*«, (39) 



>/,=la'c6' i ' m!/ sin 2 0Jfl+ — +-J-) " -— f 1 +— )«" m(r " fl) L • («) 

 r ■ (\ ma m'a*) r ma \ mrj J 



p = ±pacu jWl + — + __ I e*' 1 "" cos - (41) 



m \ ma m'ar) r 



17. The symbolical equations (40), (4l) contain the solution of the problem, the motion 

 of the sphere being defined by the symbolical equation (39). If we wish to exhibit the 

 actual results by means of real quantities alone, we have only to put the right-hand members 





