426 Mr. 8. R. Cook on the Distribution of 



to the constant momentum. Since K= ^, is always positive 



K is always positive, and ^7^ tends to decrease, i. e, when 



moving in line o£ centres, the spheres tend to repel each 

 other. When the two spheres are moving perpendicular to 

 the line o£ their centres, Hicks finds that for a perfect fluid 

 the spheres tend to attract each other. 



Koenig^ solving the same problem finds that the mutual 

 forces between two spheres moving in a perfect fluid are 



X= — ^— ^-^ sm ^(1 — cos-^) . . . (0) 



2=1 _ - ^ ^ cos d{f) — cos-^} ... (6) 



Y=0 



where a and h are the radii, c the distance apart, and the 

 angle which the line of centres makes wdth the direction of 

 motion^ Y vanishing on account of symmetry. 



When ^= -— , n being an integer, 



when 6 = n7r, 



^^ q7Tpabu\ ^ ^^. 



^= ? ^ ^^^ 



giving repulsion parallel and attraction perpendicular to the 

 stream-lines. 



As these results have been obtained on the assumption that 

 the medium is a perfect fluid, it is not possible to obtain ex- 

 perimental data to test their validity. All known fluids are 

 susceptible to changes of density, and possess internal friction. 

 The kinetic energy of a system moving in them may, ac- 

 cordingly, be transferred to the medium itself, thereby 

 necessitating the introduction of a term in the equation of 

 motion that will represent this transfer of kinetic energy. 



On the condition that the velocity of the sphere is small so 

 that the square of the velocity may be neglected, Stokes first 

 obtained the solution for a sphere in a viscous fluid in terms 

 of the potential t 



^=-iv|l-K+^J) (9) 



* Wied. Ann. Band xlii. pp. 356, 549 ; Band xliii. p. 43. 

 t Camb. Trans, ix. p. 8 (1850) ; Math, and Physical Papers, vol. iii. 

 p. 56. 



