Uniform Motion of a Sphere through a Viscous Fluid. 113 



the distribution of velocity may be supposed to tend as U is 

 diminished indefinitely. 



It is known *, however, that results obtained in this way 

 will accurately satisfy the equations, provided these be 

 modified by the introduction of constraining forces 



"K = W7] — v£, Y=u£—w!;, Z = vg—U7], . . (2) 

 . where 



fc __ Bitf Bv _ Bw B^ Bi> 3^ . 



^ _ B</~B^ V --dz~^v' p " Bar ""By- ' W 



These forces have a resultant R which is normal both to 

 the stream-line and to the vortex-line, and whose magnitude 

 is in the present case 



R=?», (4) 



where 



? = v /(^+^+^) ) »=v'(r+i 7 2 +n. . . (5) 



The magnitude of these hypothetical forces, as compared 

 with the viscous forces 



vy 2 u, vy 2 v, v\7 2 to, (6) 



where v is the kinematic viscosity, gives an indication of the 

 degree of approximation which is attained in formulae such 

 as (1). 



Now from (1) we find 



c. n STJaz . 3Ua?/ >_. 



f= ' "=s-?o ?= -2^' • • • ( 7 > 



so that for large values of r 



X-0, Y_--_ 3-, Z=--__. . . (fc) 



For the viscous forces (6) we find 



3 TT B 2 1 3 TJ B 2 1 3 TT B 2 1 



2 B<£ r 2 B^B?/ y* 2 B^B-z r K ' 



The ratio of the former to the latter is ultimately of the 

 order JJrjv, which increases indefinitely with ?•, however 

 small U may be. This is, under a slightly different form, 

 the objection which Prof. Oseen raises to the validity of (1) 



* Kayleigh, Phil. Mag. [4] vol. xxxvi. p. 354 (1893) ; Sci. Papers 

 vol. iv. p. 78. The pressure must be supposed altered by a term, 

 |p(w 2 +v 2 +^ a ), which vanishes however at the surface of the sphere, on 

 the hypothesis of no slipping. 



Phil. Mag. S. G. Vol. 21. No. 121. Jan. 1911. I 



