52 ON THE ALGEBRAICAL SOLUTION OF 



or on expansion 

 (# 3 x^ x 2 3 # 3 3 4 3 ) 



+ 3(x -K z -x i ^-x 2[ ^-x a ^)r=0 (4') 



To make the coefficient of r vanish we may take 



*V= (^1+^2+^3) A 2 (5') 



and equation (4') is then satisfied by taking 



Substituting the value of x from (5') in (6') we derive 





Now put A / = 

 and equations (3) take the form 



+3x(X 2 o; 1 + v.*x 2 + 

 A 7 Pj= A '(0^+ X) = 



(9 7 ) 



As before (9') is the integralised form of the algebraical 

 solution of (I'), and presents the roots as rational functions of 

 six variables x l9 x 2 , x a , x& X, p. ; the roots being of the third 

 degree in x l9 x 29 x s , # 4 , the ninth in x l9 x 2 , x 3 , # 4 , fx, and the 

 tenth in x l9 x 2 , x s , # 4 , X, [i. 



As a numerical example x 1 ==x 2 =x s =x i \=[L=l gives 

 49 3 =47 3 +24 3 +l 3 +l 3 . 



(iii) In general, the assumptions 1 



1 Cf . Mathematics from the Educational Times, New Series, vol. iv. No. 15225. 



