602 - 8 - 



If Q i s at distance r from the centre then 



d^ 



^ 



dt' 



(13) 



so that 



(P-), 



^•(t-|) \^_rj^ 



dS 



(W) 



From Figure u we have the following relation between R, rj and r^ 



r^ = rj^ ♦ R^ - 2 R r^ cos a 



(15) 



Before simplifying the integral in (lu) and the resulting integral in (11) we shall 

 put the equations in non-dimensional form using the following suBsti tutions:- 



Pj KP' 



T) = h^i 



P' F [i) 



n = — N 

 2a 



p h. c 



and 



P' = 



R = aX 



dS = a dS' 



(16) 



where 



1 



= 1 + 



256 o a 



h 10 5 rr |0i 



F.qu'tion (lU) then becomes 



p, KP- on, c^ 



16 



u 77 a 



i- (T-^) fl - Xj'^) 



dS' 



(17) 



i' (T-^) (1- x^'^ 



ds- 



(18 



in which i is regarded as a function of T and the surface integral is taken over a circle of 

 unit radius (Figure 5). Also frm (11) and (16) the equation of motion Becomes 



d^l . r (J) = ^ .-NT 



ii 



(19) 



I = 2 TT 



P- (x^, T) (1 - x^^) x^ 



dx, 



(20) 



and P' is givon by oqu-ition (13). 



Finally the non-dimensional fom of equation (15) is 



x,^ + x^ - 2 XXj cos a 



(21) 



