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 lt X 2 , . . ., X n )=0 (1) 



be a homogeneous indeterminate cubic in n variables X lt X 2 , 

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

 X 1 =a lt X 2 =a 2 , . . ., X n =a n) (2) 



so that <(!, 2 , . . ., a n )=0 (3) 



where by hypothesis a lt 2 , . . . a n do not all vanish. 



Now make the substitutions 



X^Xjr+a^ X 2 =x 2 r+a 2 , . . ., X n =x n r+a n (4) 



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



A^+Atft+Atf+^a^ a 2 , . . . a n )=0, (5) 



where A& A 2 , A are homogeneous integral functions of 

 x l9 x 2 , . . ., 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^O (6) 



which being linear and homogeneous in x lt x 2 , . . . x n can 

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

 terms of x l9 x 2 , . . . x n _! be substituted in A 2 and A 3 , 

 which will in general be finite functions of x l9 x 2 , . . ., x % . v 

 The equation (5) is then identically satisfied by taking 



r=-A'JA'i (7) 



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

 terms of x l9 x 2 , . . ., x n _ l by (6). The values of x n and r, given 



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