446 



3 0, 



of two factors, one the "attraction coefficient " ,.^^2, which is a geometrical factor, and the other 



ox 

 an integral expression involving the radial motion of the bubble and independent of the geometry of 

 the surfaces. 



The Velocity Potential Equations . 



The most suitable co-ordinates in which to solve the problem are oblate spheroidal 

 co-ordinates r, s, i/', as shown in Figure 1. AB is the disc, radius c, and A, B, are the foci of 

 the confocal oblat;! spheroids, r = constant and the hyperboloids of one sheet s = constant. OP Is 

 the axis of symmetry, and the explosion centre is at X on this axis, distance d from the disc. It 

 is convenient to choose o -^ r ■$ =>, so that s varies between - 1 and + 1. This choice mal<es the s 

 co-ordinate continuous in the region of the field. It Is to be noted that s changes sign on passing 

 through the disc. 



The relationship between the spheroidal co-ordinates r, s, and cylindrical co-ordinates x, p, 

 origin at o and the x co-ordinate positvc in the direction of OX, is 



X = c r s (1) 



p ^ c [(!♦ r2)(i- s^)]* (*) 



Since the problem is entirely symmetrical with regard to i// it will not appear in the equations. 



The disc AB is thus the surface r = o. Denoting the potential due to the point charge at 

 X (r = r. , s = I. i// = o) by cp^ we have* 



*1 = ^£ '^^ ^' «n ''^o' ''n t'^> "n '^' ' ^ 'o ^^^^ 



0, = i £ (2 + 1) P (ir ) 0^ (ir) P (s) r > r (jb) 



1 c 1^.^ n n u n M o 



Where P and are the Legendre functions of the first and second kind, and i = /-I. 



To this must be added a potential ip, which satisfies Laplace's equation, vanishes at infinity, 

 and has no singularities in the region of the field. Thus 



cCj = 5^ A P (s) (ir) (U) 



n=o 



The potential <p = (p. "*" <p2 ^^s to satisfy the condition that at the disc, i.e. the surface r = o, 

 its gradient normal to the surface is zero, since the disc is both rigid and fixed. This gives an 

 equation from which the coefficients A may be determined. Thus 



I {An "^n' 'i--"' * r '2" * ^' ^n ' ' ''o' '"n^ (!-"» ''n '=' = ° <5) 



for r = 



(2n + 1) 0_ (lr„) 



(it is 



Static and Dynamic Electricity. Smythe, McGraw Hill 1939, p. 165. It nay be verified that 

 the potentials given in (3a) and (3b) satisfy Laplace's equation In these co-ordinates, and 

 reduce to H.vino's expansion of ^.j^ j^ ) , or ^^^_^ -^ respectively, along the axis of symmetry 



s = 1 (see Whittaker and Watson, Uth Edition p. 322). 



