46 Transactions of the South African Philosophical Society. 

 This can be simplified still further by putting — 



We now have- 



K r = -jJi (excluding r = 0. 



L o =0 

 L x = l 

 L 2 =l 



L 3 = (L 2 L I + L I L 2 ) 



L n = (L^Lj + L n _ 2 L 2 . . . + L 2 L„_ 2 + L^L^) 



To determine the L's put — 



z = L 2 + L 3 # + L 4 ic 2 4- . . . 

 so that 1-\-xz = ~L-\-Ij 2 x-\-Ij 3 x 2 +... 



x 2 (1-\-x 2 ) 2 = (L q + 1 L 1 x-\-Ij 2 x 2 ...) 2 



= L^+a»(L L I + L I L )+^(L L 2 4-L I .L 1 + L 2 L )+.... 



= L 2 # 2 + L 3 cc3 -}- L 3 £ 4 

 =# 2 £ 



.*. (l + # 2 ) 2 = 2 



,-. z 2 x 2 + 2 (2x-l) + l = 



1 _ ^2 — /l — 4# 

 ."• z= —^ , selecting the appropriate sign 



_1 , 1 ,1 2 , 



X 



where c r is the coefficient of - x r in the expansion of (1 - 4#) 2 . 

 Hence L r = -c r . 



A 



Now the expansion of (1 — \.xf in ascending powers of x is conver- 



1 c L 



gent if x <-.; hence L-^rM, and .\ L -y^"-^»4. 



n=x v n 



n—ao -*-"/j 



Thus the solution of the differential equation is — 



a 7 X „ . a"X 2 



TT 



where L-f^- 1 ^ 4. 



.[V <2 2 A „ a;A 2 1 



\f-«i* 1 + — -E + — H r .. 

 L a x a; ° J 



This will therefore be convergent if 



a 2 X 

 a, 



< 



