1879.] I^a^sait'??^ spheres in a fluid. 279 



8. Passing on now to consider the force on B towards A, let 

 (f) denote the velocity-potential, determined by the successive mass- 

 images. The pressure at any point, taking the density of the fluid 

 for unity, is given by 



p = const. - g^ - i F', 



and the force on B from A is 



X= i pcosd dS, 



the integral being taken over the sphere, and the pole of being 

 BA. We know that V^ has no effect on the resultant force, 



X = - 2'7rb' r ?^ cos d sin 6 dO. 



dt 



Now ^ is made up of a sum of velocity-potentials due to single 

 sources and sinks. Let ii be one of these at a distance r from B. 



I n 



Then the part of X due to this is 



^ 12 d f"" yu.„ cos 6 sin 6 dO 

 dtJo JF+r' - 2hr cos (9 ' 



the differentiation not extending to b. 

 It can easily be shown that this is 



according as fi^ is outside or inside B. 



So if c be the distance between the centres of the spheres and 

 p^ be the distance of the source of a mass-image in A from A, 

 and pn the distance of the other extremity, the part X^ of X due 

 to this mass-image is 



_ 27rh^ d fjb^ 2'7rb^ d f P" fi„dx 



3 dt'(c-p„y 3 'dtjp„'{p„~pj){c~xy 



_2'jr¥d (p„-p„>. 

 3 dt{c-pj{c-p:)' 



and the part of X due to mass-image v^ within B, cr^, cr^ denoting 

 distances from B is 



■y , _27r d . , 2'Tr d 1'°'" vjcdx 



^^" ~irdt ^""-"^''^ ~Ydt],rcr^ -^7 



IT d , ,. 



