1879.] 



where 

 M=h 





1 + 



pulsating spheres in a fluid. 

 {r-a:'){r'-a'-by 



281 



+ 



{{r' - dj - 6Vj [ir' - ay + {r' - b'f - (r* - a'b')YJ ' 



r r' aW " 



dt l{r' - by "^ (r' - a' - b') {{r' - by - a'ry_ ' 

 So intercliangmg a, b, the force on A from B due to the pulsa- 



N=Fa^ 



tions of B is 



and on jB from A is 



~W'~dt' 



V_ dN^ 



4r' dt ' 



where M^, N^ are M^ . N^ with a.b &c, interchanged. Finally then 

 when both pulsate the force on A from B is 



and on B from A is 



_a^(dM, Idm 

 ' 4>r''\dt '^r dt}' 



l_{dM_ 1dm 



^ 4!r'\dt "^r dt)' 



( N\ ( N\ 

 or writing /i,^, yu^ for \ [M^ + ~ ) » 4 ( ^^2 + "J respectively, 



' r' dt ' 



F = 



I, 

 r . 



r'' dt' 



These are the forces at any instant, the spheres being supposed 

 held fixed. To determine the mean forces when the pulsations 



are quick, we have to find the mean values of a^ —y- and b^ -^ . 



If one is not pulsating, as for instance B, then since b is 



constant, the mean value of b^ -^ is zero. Also M^ = 0, and the 



mean force on A is of the order of the inverse cube of the distance. 



5. If we confine our attention to that part of the forces 

 depending on the inverse square of the distances, 



a dm. jodb 



4 dt 



dt' 



b dm, 07 da 



^^=l-dt='^''^dt' 



