52 ON THE ALGEBRAICAL SOLUTION OF 



or on expansion 



(# 3 xf x 2 3 x a 3 # 4 3 )r 3 +3(o; 2 X o^X x 2 2 [i.+x 3 z ^)r 2 



+ 3(a; X 2 -:r 1 X 2 -cc 2 [A 2 -a: 3 !A 2 )r=0 (4') 



To make the coefficient of r vanish we may take 



z = (X%+ 1^2+ f^ag/X 2 (5') 



and equation (4') is then satisfied by taking 



r=3(X* 1 2 + [^ 2 2 - [LX3 *-ix ( ?)l(x < ?-x ] 3 -x z s -x ! ?-x t 3 ) (6') 



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



Now put A'=(X 2 cc 1 +jA 2 a; 2 +(x 2 a; 3 ) 3 X 6 (x 1 3 +x 2 3 +x 3 3 +a: 4 3 ) (8') 

 and equations (3) take the form 



A / P =A'(a; r+X)=X(X 2 a; 1 - 

 + 3X(X 2 z 1 + (A 2 a: 2 + (i^X 4 ^ 8 - 



A l-f . A /'V "j*_L ^i \ ~\ \~\ 2/v i 1 1 2/v* i , . 2 v \3 "i 7 //>* 3 [ /y* 3_j_ /v o 1 /> Q\ 

 * 1 * * V**''! * ~T~ / ** \ " t*/i |~ wt. X/o~j~ JX *^"l/ " V 1 l^ *^2 1^ *vo ] *X/ 7 



AT 2 = A'( 



A'P 3 = A'(a; 8 r- pi)= - f i(X 2 x 1 + iA 2 + !. 2 a: 3 ) 3 + X 



(9') 



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

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

 six variables x lt x z , x s , x t , X, JA ; the roots being of the third 

 degree in x lf x 2 , x a , x^ the ninth in x v x z , x s , x t , (x, and the 

 tenth in x v x z , x 3 , x, X, [>.. 



As a numerical example Xj=a; 2 =a; 3 =a; 4 =X=(jL=l gives 

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



(iii) In general, the assumptions 1 



1 Of. Mathetnatics from the Ediu-ational Times, Now Series, vol. iv. No. 15225. 



