50 ON THE ALGEBRAICAL SOLUTION OF 



these as before may be integralised if the equation be homo- 

 geneous. On the other hand, if <=0 be an indeterminate 

 biquadratic, in at least four unknowns, then equation (5) will 

 contain an additional term Ar* where A is homogeneous and 

 of the fourth degree in the unknowns. When equation (6) 

 is solved and the value of one of the variables so determined 

 is substituted in A 2 , A' 2 =Q becomes an indeterminate quad- 

 ratic in at least three unknowns. If, therefore, rational 

 solutions of A 2 =0 can be found, equation (5) is solved by 

 taking r= A 3 /A^ and it is clear that in general the solutions 

 of the final equation <=0 will be numerical or algebraical 

 according as those of A 2 are algebraical or numerical. It is, 

 however, exceptional for the subsidiary equation A 2 =0 to 

 yield rational solutions. 1 



6. The above convenient method for the solution of 

 indeterminate cubics will be illustrated by some typical 

 examples. From these it will appear that there are few 

 problems, if any, in indeterminate cubics to which it is in- 

 applicable, or in which the results furnished are less general 

 than those of another process. 



QUESTION 1. Solve algebraically the equation 



P 3 =Pl 3 +P 2 3 + +Pn* (A) 



(i) Let n=3 t so that we have to solve 



P 3 =P 1 3 +P2 3 +P 3 3 (1) 



Here we may take as our particular solution 



P =P 1 =X, P 2 =-P 3 =[* (2) 



Making then the substitutions 



P =V+ X Pi=ay+X, P 2 =# 2 r+fx, P s =x 3 r-v., (3) 



equation (1) takes the form 



( iy+ X) 3 = ( Xl r+ X) 3 + (x 2 r+ (x) 3 + (x 3 r- fx) 3 

 or, on expansion 



[ x 2 a; 2 -(jL 2 a; 3 )r=0. (4) 



1 See the writer's paper: 'On the Algebraical Solution of the Indeterminate Equation 

 XX 4 + n Y 4 = vZ^ + p V* ' : in course of preparation. 



