Motion of a Sphere through a Viscous Fluid. 121 



Bence if we put 



<f> = A logr + A 1 ^ 1 ogr + ..., . . . (53) 



we find that the conditions u= — U, v = 0, z# = 0, will be 

 satisfied for r=a, provided 



C = 2U/(i- 7 -logPa), A =-C/2/<r, A^iCa 2 , (54) 

 approximately. Hence near the sphere we have 



1 



«=|c| 7 -^-flog^T+^(^-^)|^ 2 logr}, 



v = -C(r*-a 2 )^Kr\o«r. 

 The vorticity is given by 



(55) 



which for large values of kr takes the form 



^-<\Z(wy- x) ■ ■ ■ -./**> 



The general interpretation would follow the same lines as 

 in the case of the sphere. 



To calculate the force exerted by the fluid on the cylinder 

 we have to integrate the expression 



or 





-pl+^+p^+y \ . . . (58) 



P UA 



with respect to Ihe angular coordinate (#) from to 2ir. 

 The products of plane harmonics of different orders will of 

 course disappear in this process. The first term of (oS>) 

 gives, when r is put equal to a, 



" ccs' 2 6d0=- TrpJ] A = TTfjuC, . (59) 







by (45), (53), (54). The second term contributes, on 

 substitution from (55), ir/nC. The third term vanishes 

 identically, to our order of approximation. The final value 

 for the resistance per unit length is therefore 



M* i 4 T U az S (60) 



i-7-log(^a) 



The investigation is subject as before to the condition 

 that ka, or Ua/2r ? is to be small. 



