706 Lord Rayleigh on the Motion of 



of the functions is found to depend upon the ratio of densities 

 of the sphere and liquid. This may be rather unexpected ; 

 but I am unable to follow Mr. Basset in regarding it as 

 an objection to the usual approximate equations of viscous 

 motion. 



§ 6. We will now endeavour to apply a similar method to 

 Stokes's solution for a cylinder oscillating transversely in a 

 viscous fluid. If the radius be a and the velocity V be 

 expressed by Y = Yn,e int , Stokes finds for the force 



Y=-WinN n e int (k-ik ! ) (35) 



In (35) M' is the mass of the fluid displaced ; k and k' are 

 certain functions of m, where m = iaV (njv), which are 

 tabulated in § 37. The cylinder is much less amenable to 

 mathematical treatment than the sphere, and we shall limit 

 ourselves to the case where, all being initially at rest, the 

 cylinder is started with unit velocity which is afterwards 

 steadily maintained. 



The velocity V of the cylinder, which is to be zero when t 

 is negative and unity when t is positive, may be expressed 



by 



(36) 



t- t . 1 f" sin nt 7 

 V = i+-1 dn, 



in which the second term may be regarded as the real 

 part of 



i r°° p int 



-A —dn (37) 



We shall see further below, and may anticipate from Stokes's 

 result relating to uniform motion of the cylinder, that the 

 first term of (36) contributes nothing to F; so that we may 

 take 



7T L 



e in* (k-iti)dn, 



corresponding to (37). Discarding the imaginary part, we 

 get, corresponding to (36), 



M' C m 

 F= 1 (k cos nt + k' sin nt)dn, . . (38) 



77 Jo 



Since /j, k' are known functions of m, or (a and v being 

 given) of n, (38) may be calculated by quadratures for any 

 prescribed value of t. 



It appears from the tables that k, ¥ are positive throughout. 



