- 3 - 447 



(it is to be noted that the function Q^ has a branch line along the real axis between -1 

 and +1, and that the value of Q^^ (ir) at r = o is really the limit of 0^ (iS), as 8 tends to zero 

 through positive values). 



The desired potential due tc the "induced" images is therefore 



^^ = -\ I (^*3)Q,,*l(-oN3„>:(-'p,,^:'^' (*' 



m-o 



The "attraction coefficient" of the disc is '^2 = - '^2 at s = 1 and r = r^ = -, i.e. 



3 X 3 r 



at the explosion centre X. 



l^\ = -ii 2 («m . 3) 0,„ * it'^o' CL .- if'^c' (7) 



'2\ = -11 



: J at X 77 c 



Equation (7) may be put in finite form as follows. Heine's development of l/t-z is: 

 (Whittatier and Watson, p. 321) 



T=z 



1 (2n + 1) P„ (z) (t) (8) 



The series in (8) is valtd for all points z lying inside an ellipse in the complex plane 

 passing through t and with foci at ±1. Changing the sign of z throughout and subtracting this new 

 expression from (8) gives 



£(^.3)P,,.t(^>P,.t'')=^-,- 



DivTding both sides by -2 (t - z) , and integrating both sides with respect to z from -1 

 to +1, yields 



I (^.3)P,^l(",,^,^I<" --U] 



1 (t^- z')(t - z)' 



^ '°{r^)-irf-|rT-M^ (') 



Since the reversal of the order of integrating and summing may be justified for all values of 

 t not on the branch line of the Q functions, i.e. the portion of the real axis between -1 and +1. 



Finally, replacing t In (9) by (ir ), and taking only the principal value of the complex 

 logarithm in (9) as necessitated by the boundary condition at the surface of the disc, equation (7) 

 for the "attraction coefficient" of the fixed rigid disc becomes 



3 X Jat X 77 c' J 2r ^ Ur 



(10) 



^r^ (1 + r ) 

 o o 



where r = d/c 



It may be verified that as d tends to zero, the attraction coefficient In (lO) tends to -l/4d 

 the attraction coefficient for an Infinite rigid plane. Recapitulating the result obtained in R.R.L. 

 Note ADM/210/ARe, the velocity u of the bubble towards the disc is therefore 



')... {"'-v i\ ''■''''] 



^ ) } 3a'a - -°, J a' J' dt \ (n) 



Where a is the radius of the bubble at tfme t, and "d cb^ri x Is given by (lO). 



Comparison 



