309] OSCILLATIONS OF A SPHERE. 569 



and by comparison with (x), it appears that this must involve surface 

 harmonics of the first order only. We therefore put w = l, and assume 



x =j5^ (xii). 



Hence U=T*~ -* 5 + 



The conditions (x) are therefore satisfied if 



= u .................. (xiv). 



The character of the motion, which is evidently symmetrical about the 

 axis of X, can be most concisely expressed by means of the stream-function 

 (Art. 93). From (xi) or (xiii) we find 



...(xv), 

 or, substituting from (3), 



If we put .2?=rcos0, this leads, in the notation, and on the convention as 

 to sign, of Art. 93 to 



Writing u = ae i(a * +e) ................................. (xviii), 



and therefore h = (l- 1) /3, where /3 = (cr/2i)*, we find, on rejecting the imaginary 

 part of (xvii), 



- J^ { 



cos {ir<-^r- a) + e} 



At a sufficient distance from the sphere, the part of the disturbance which 

 is expressed by the terms in the first line of this expression is predominant. 

 This part is irrotational, and differs only in amplitude and phase from the 

 motion produced by a sphere oscillating in a frictionless liquid (Arts. 91, 95). 

 The terms in the second line are of the type we have already met with in the 

 case of laminar motion (Art. 298). 



To calculate the resultant force (X) on the sphere, we have recourse to 

 Art. 306 (18). Substituting from (xii), arid rejecting all but the constant 



