INDETERMINATE CUBIC EQUATIONS 53 



+ (x 2m r+p) 3 + (x 2m+l r-p)* 



are sufficient to give algebraical solutions of equation (A) for 

 the two cases n=2m and n=2m+l. In the former case the 

 roots will be functions of the 2m quantities x l9 x 2 , . . ., x 2m , 

 and of the m quantities X, jx, . . ., p ; in the latter the roots 

 will be functions of the 2m+l quantities x lf x 2 , . . ., x 2m+l , 

 and of the m+ 1 quantities X, jx, . . ., p. 



QUESTION 2. Solve the equation 



xy(x-y)=\z* (1) 



knowing a particular solution, say x=a, y=b, z=c so that 



&(-&) =Xc 3 (2) 



Put x=x 1 r+ a, y=x 2 r+b, z=x B r+c, (3) 



and (1) on expansion and rearrangement becomes 



xx 



+ (ab x 1 x 2 +ab ax^+bx^ 3Xc 2 ic 3 )r=0 (4) 



Hence, making the coefficient of r vanish by taking 



x s =(ab x 1 x 2 +ab oZg+fce^/SXc?, (5) 



equation (4) is satisfied by taking 





CCg X-iXnVXi 



, - 6)(q 2 - ab + 6 z )6x 



6 3 (2a - 6) V - 3 



on substituting the value of x 3 given by (5) and replacing Xc 3 

 by ab (ab) from (2). 



Hence, if we call the denominator of r, A, we find, finally, 



(B) 



^y=(x 2 r+b)=b(2ab) 3 (bx 1 ax 2 ) 3 



&z=(x 3 r+c)=-(a+b)(2a-b)(2b-a)(bx 1 -ax 2 ) 3 

 which, it is to be remarked, is not an algebraical solution, but 



