- 11 - 155 



The recurrence relation between the coefficients in (50) Is found to be 

 (r*l)(2rtl) ^>^r*i. ^ '~ '"'''* '""'' ^^ *r * ^" (''-l' (^''-l) P * (rv-l){9+3 cr) } A^_j 



- (r-i)(r-l) }0a k'«^_j - (r-3)(2r-3) 6 cr k «^_j - 2 (r-4)(r-2) cr A^_^, (6l) 



•here k = 1 or a as the case nay be, and 



k = 1 '" " °-m 



i « 3(2 o- k"- (3 ta) k ♦ 2) = 3(o--i) = 3(3 a^ ♦ or a^ - 4) 



m = i(30O- k" - (la ♦ 6cr) k ♦ 6) = 3(2 C - l) = 2 i ♦ 3 (42) 



20 cr k' - 3 - cr 



»tr - 3 



2 O- a' - 3 - a- 



By taking r any positive integer (including zero) In (5l) and taking B_^ - for r > 0, we obtain 

 the coefficients of the first series in (50), and by taking r eqjal to half an odd integer we obtain 

 the coefficients of the second series. 



It will be noted that the approximations to /3 near a » 1 and near a " a are respectively 



4„ {l ♦ (rh-l) (a - l)} + \ ,, /(a - l) 



1/2 



/3 = A.^{l. 



(a„ 



> * *'wy(a„-a) 



1/2' '"m 



(53) 

 (54) 



In the numerical cace worked out (corresponding to a = 30.732) it *as found that the above 

 series were only suitable for calculation in the regions a = 1 to 1.5 when z * a ~ I, and a = 10 

 to 30.73 when z = a^ - a*. Hence to supplement these series, the Taylor expansion 



C.2 + C,z' 



(55) 



of the solution, relative to any ordirary point z » k, was obtained. The recurrence relation for 

 the coefficients of this expansion is 



(r+2)(r»l)(+hk^) C^^^ = (rn) (2rn) (-( k) C^^^ + {-Vmr^ + („.i) q} c^ 



+ (r-l)(2r-l)(-p) + (n-l)(9+3CT) C^.j + (r-2) (r-i) (-30a k^) C^,^ 



+ (r-3)(2r^3)(-«o- k) C^_^ * (r-4) (r-2) (-2 cr) C^_^ 



where 



h = 2crk'*-(6t2cr)k+6 



q = (9 + 3 0-) k - 12 



and ' , m and p are g Iven in (52). 



The relation between. /3^ and /5 , 



The above expressions give/5^ ^ as a function of a (with two arbitrary constants) over a ny 

 half-period. If s Is odd, /3 corresponds to an expanding phase, and c+i '° '''^ succeeding 

 contracting phase. To obtain the relation between these /3's we have the conditions that /9 and p 

 are continuous at a = a^. Ttie value of /5 is given by 



^-.i^i.tSj 1^ / {a-} . a-' . U ia^ . ,)} (58) 



da a a A ^ 



(56) 



(57) 



dj 

 d a 



/ (a - a.) 4. (a), say. 



and the positive or negative value must be taken according as we are considering an expanding or 

 contracting phase. 



Hence 



