4 Lord Rayleigh on the Flow of 



Thus- . . 



± TT r c 2 , 13UV1 ,^ UV _ 



UV f ,/ c 2 , c 4 \ cos 301 , 1K . 



satisfies all the conditions and is the value of <f> complete to 

 the second approximation. 



That the motion determined by (15) gives rise to no 

 resultant force in the direction of the stream is easily 

 verified. The pressure at any point is a function of q 2 , and 

 on the surface of the cylinder q 2 = c~ 2 (dcf)/d0) 2 . Now 

 (tty/dO) 2 involves in the forms sin 2 0, sin 2 30, sin sin 30, 

 and none of these are changed by the substitution of it — 9 

 for ; the pressures on the cylinder accordingly constitute 

 a balancing system. 



There is no particular difficulty in pursuing the approxi- 

 mation so as to include terms involving the square and 

 higher powers of U 2 /a 2 . The right-hand member of (6) 

 will continue to include only terms in the cosines of odd 

 multiples of with coefficients which are simple powers of r, 

 so that the integration can be effected as in (11), (12). And 

 the general conclusion that there is no resultant force upon 

 he cylinder remains undisturbed. 



The corresponding problem for the sphere is a little more 

 complicated, but it may be treated upon the same lines with 

 use of Legendre's functions P w (cos 0) in place of cosines of 

 multiples of 0. In terms of the usual polar coordinates 

 (r, 0, o>), the last of which does not appear, the first approxi- 

 mation, as for an incompressible fluid, is 



* = Ucostf(r+^) = u(r+^)p 1 , . . (16) 



c denoting the radius of the sphere. As in (8), 



d^d£_d$d£ 1 d$d£_ {J , r/_ 36c 6 9r>\ 

 Z dx dx ~ dr dr + r 2 d6 dd \\ W + 2r 10 ) * 



, (W 24c 6 , ,V\ P 1 



on substitution from (16) of the values of <f> and q 2 . This- 

 gives us the right-hand member of (6). 



