464 Transactions of the South African Philosophical Society. 



than that of the corresponding E where the set of E's is given by 

 the equations — 



Write — 



E o =0 



E 3 = - . 2E X E 2 + p- (3E E 3 + 6E3A) + • • 

 cii Aa~i 



&c 



,, = E I ^ + E^ + E 3 ^+... 



.(17) 



a, 



Then - . tf + o~ • n 3 + « • >7 4 + 



a, 



=- 3 (E I 4 + E 2 ^ + ...) z + o- 1 -(B I | + E 2 r+...)3 + ... 



= »7 " Bj4 



= »-£ 



s. a, , a, a. . 



• '. 4 = v ~ — • T - c> - ^ ~ q • ^ ■ • • 



a x ^a r oai 



The reversion of this series gives an expression for r\ in ascending 

 powers of £, the coefficients therefore being the E's. 



In a previous paper * I have shown that the reversed series is 

 convergent if |£j<J, where J= x /(l + 2/3) 2 + l —(1 + 2/3), ft being 



the greatest of the set — ^- —-... 



a x ' 2ai' oai 



Hence finally the solution of — 



dy 

 y= ai y + a 2 y + a 3 y> + . . . 



with initial condition y — \ when t — Q can certainly be obtained by 

 successive approximation in a convergent series of powers of \ pro- 

 vided 



a x is negative, 



and | X | < Jl + 2/3) 2 + 1 - (1 + 2/3), 

 where ft is the greatest of the set 



* Brit. Assoc. Eeport, 1905, p. 319. 



a 3 

 2a* 



a^ 

 3<2t 



