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XV. On the Uwform Motion of a Sphere through a Viscous 

 Fluid. B'j Prof. Horace Lamb, F.H.S.* 



A N important, contribution to tins subject has recently 

 JTjl. been made by Prof. 0. W. Oseenf, of Upsala, who 

 calls attention to a certain limitation affecting the validity of 

 the accepted solution J, however small the velocity may be, 

 as regards points at a sufficient distance from the sphere. 

 He proceeds to give an amended solution which appears to 

 be free from this defect, whilst it gives the same distribution 

 of velocity in the immediate neighbourhood of the sphere, 

 and consequently the same value of the resistance, as the 

 older theory. 



Prof. Oseen's analysis appears to have more than the 

 immediate object in view, and is for the present purpose 

 somewhat long and intricate. Considering the great interest 

 of the question, it may be permissible to indicate a shorter 

 way of arriving at his results, and to add a few comments 

 dealing somewhat more explicitly than he has done with 

 their interpretation, and with the estimation of the degree 

 of approximation which is attained in different parts of the 

 field. 



The problem is most conveniently treated as one of 

 " steady" motion, the fluid being supposed to flow with the 

 general velocity U, say, parallel to x. past a fixed spherical 

 obstacle whose centre is at the origin. The distribution of 

 velocity is then given, on Stokes's theory, by the formulae 



a\ 1 XT , , 9X B 2 1 



-Ufl-^-iu^-a^^ 



1 rl 2 1 



4 v ' dx^zr* 



(1) 



where a is the radius of the sphere, and r denotes distance 

 from the centre. These formulae are based on the assumption 

 that the inertia terms in the hydrodynamical equations, which 

 are of the second order in u, v, w, may be neglected. They 

 were, in fact, only propounded as a limiting form to which 



* Communicated by the Author. 



t " Ueber die Stokes'sche Formel, und liber eine verwandte Aufg-abe 

 in der Hydrodynamik," Arkiv for matematik, astronomi og fysik, Bel. 6, 

 no. 29 (1910). 



% Stokes, Camb. Trans, vol. ix. (1851) ; Sci. Papers, yoL iii. p. 55. 



