ON THE ALGEBRAICAL SOLUTION OF IN- 

 DETERMINATE CUBIC EQUATIONS 



PART I 



1. Theorem. If a particular non-zero solution of a 

 homogeneous indeterminate cubic equation be known, then 

 an algebraical solution can in general be found. 



Let <t>(X v X z , . . ., X n )=Q (1) 



be a homogeneous indeterminate cubic in n variables X lt X z , 

 . . ., X n , and let it have a particular non-zero solution, say 

 Jf 1 =a 1 , X z =a z , . . ., A n =a B , (2) 



so that <(!, a z , . . ., a n )=0 (3) 



where by hypothesis a lt a z , . . . a n do not all vanish. 



Now make the substitutions 



X^Xjr+a^ X z =x z r+a z , . . ., X n =x n r+a n (4) 



and equation (1) becomes on expansion in powers of r 



A^+A^+Ajr+^a^ a 2 , . . . aj=0, (5) 



where A 3 , A z , A 1 are homogeneous integral functions of 

 x lt x z , . . ., x n of the third, second, and first degree, 

 respectively. 



The term in equation (5) independent of r vanishes by (3). 

 The coefficient of r can be made to vanish by solving the 

 equation A^Q (6) 



which being linear and homogeneous in x lt x z , . . . x n can 

 always be solved. Let the value so found for x n say in 

 terms of x lt x z , . . . x n , 1 be substituted in A z and A 3 , 

 which will in general be finite functions of x lt x 2 , . . ., n -i 

 The equation (5) is then identically satisfied by taking 



r=-A'tfA' s (7) 



where A' z , A' 3 are what A 2 , A 3 become when x n is expressed in 

 terms of x lf x z , . . ., x n _^ by (6). The values of x n and r, given 



