Motion of a Spliere through a Viscous Fluid. 119 



At points well within the wake, where .k(r — x) is small, 

 we have 



u=-l^, v = 0, w = 0, . . . (37) 



and •> ,. 3 „ 



f=0, y=r>Vka^, {= - 7) \]ka^ 2 , . . (38) 



9 ii „ 9 



approximately. These make 



X=0, Y=~U 2 ka 2 ^, Z=-^ka 2 ~ . . (39) 

 The viscous forces are found to be 



vv*u=2vkC% 9 v^ 2 v = 2vkVA- vv 2 w = 2vkC~. (40) 



approximately. The ratio of the magnitudes is of the 

 order ka. 



Near the surface of the sphere we have u—— U, v — 0, 

 w=r0, approximately, and therefore, from (2) and (19), 



X=0, Y=-u|^, Z=-u|-^, . . . (41): 



or, by (27) and (29), 



X = 0, Y=|u 2 a^ Z-~U 2 a^ 6 . . . (42) 



The resultant is therefore of the order TP/a. The viscous 

 forces are obtained from (21) and (2S) ; thus 



3 TT 3# 2 — r 2 , 3 TT ?>xy 



"V« = 2 vUa ~? — > "V*« = 9 vUa "^ ' 



W 2 ^=|vUa^, . (43) 



giving a resultant of the order vU/a 2 . The ratio of the 

 magnitudes is therefore Uajv, which has already been 

 assumed to be small. The approximation, although less 

 perfect here than on Stokes's theory, is seen to be adequate. 

 I may repeat that the object of this note has been merely to 

 give a simpler demonstration of Prof. Oseen's results, and to 

 elucidate a little more fully their scope and significance. I 

 have not touched upon another aspect of the question which 

 is referred to in his paper, and which is apparently to form the 

 subject of a further investigation. 



