46 



Proceedings of the Royal Irish Academy, 



(2) Solve the equation (x^ ax ^ h = 0. 



I _ ^ _ " 



- ia/S-" + -Aa-(3-' - R 



- hafi-'' + -i-so"^-' - R 



- ^,c^^ - R ; 



1 _3 _7 



X = {- hf - ^a{- h)' - i-.a'' {- h)"^' - R. 



AslJ-h has five values, this series also has five values, which 

 may be supposed to be the five roots of the equation. In order to 

 prove this point, let the five values of JJ- i be 7na, 7na^, 7na^, ma^, ma^, 

 where a, . . . . are the five values of 1, so that 



a + + + a* + = 0. 



Insert these values of Jj- 1 successively in the series for ^. We shall 

 thus have five series. It will be found that their sum, and the sum 

 of their products two at a time and three at a time, vanish. The sum 

 of their products four at a time = a ; and their product all together = - b 

 (see also § 21). Thus the five series are the five roots of the equation ; 

 and as these series are all contained in the original series, that series 

 is the complete algebraic, or, rather, transcendental, solution of the 

 equation 



+ ax + b = 0. 



There is one condition attached — that the series be an infinite one. 



(3) Divide the same equation by x^ and by x, and solve. This 

 process enables us to put the equation in four move forms, namely, 



5 4 1 4 ^ - 1 

 X ^ -\- jX ^ = - -, X ^ + - X"' = , 



a a 



x^ + bx~^ = - a, x~^ +y x^ = - ^ : 

 b b 



and each form can be separately solved by dividing /? by the four 

 operations 



/3-=+^/3-% Z?-*.*/?- yS'^*/3-', 13-^+1/3', 



