Shape of Large Babbles and Drops. 509 



As before* we have, very closely, 



p _ y V ia 2 — y 2 

 y/l+f" 2a 2 ' > 



which, substituted in the small term in (ii.), gives on 

 integration 



J, _i \ f (4a 2 -y 2 )# 2a 2 4a 3 ... , 



when y = q, p = co , giving 



2a 2 = Q 2_ Sa L + (4a 2 - g 2 ) 1 4a^ 



= ?2+ 4a2 2- 2 f 3 (4-V2), 

 /J, or 



putting q 2 = 2a 2 in the small term involving r. 

 If we adopt Poisson's notation, in which 



(-S)--- 



we get 



a l 2 =o 2 + 2a2 - g --^(2 V2-1). . . . (ii.6) 

 a //, or 



In most practical cases the bubble or drop may be taken 

 plane at the vertex, in which case we have 



a 1 2 = g 2 -^(2 v/2-1) 

 or 



= 2* --606^' (iii.) 



Following the second method o£ approximation of the 

 paper cited t, and assuming the bubble plane at the vertex, 

 we obtain, taking account of sign, 



d<f> 1 y 



dy r a 2 sin <£ ' 



which gives as before, proceeding by the method of variation 

 of parameters, 



7 (a 2 . c a 2 ii c \ , ,. v 



ydy=-\— sin- ;f cos- lac. . . . (lv.) 



\ r r r r ) 



* L. c. p. 841. 

 t i. c. p. 842. 



