INDETERMINATE QUARTIC EQUATIONS 77 



Hence we shall have 



(w 2 +v 2 ) 4 =(% 2 -v 2 ) 4 + (2uv)*+(x+y)*+ (x-y)*+ (2y)* (1) 

 provided 2uv(u 2 -v 2 )=x 2 +3y 2 (2) 



Now if for the moment we regard v as a constant, equal to 

 v' say, this equation may be written 



which, being a non-homogeneous indeterminate cubic in 

 u, x, y, can be solved algebraically, if a particular solution is 

 known. 1 But a particular solution of it is obviously u2, 

 v'=l, cc=3, y=l, and others are easily found, for example, 

 (u, v, x, y)=(7, 6, 3, 19), or (7, 6, 15, 17), or (7,6, 27, 11), or 

 (7, 6, 33, 1). Hence an algebraical solution may be found. 

 To solve (2) we may therefore put 



u=Xjr+u', v=v', x=x z r+x', y=x 3 r+y' (3) 



where we suppose (u', v', x', y') to be a particular solution of 

 (2), i.e. 



2ii'v'(u' 2 -v'*)=x' 2 +3y' 2 (4) 



Making the substitutions (3), equation (2) then becomes 



2(x 1 r+u')v'[(x l r+u') 2 -v' 2 ]=(x 2 r+x') 2 +3(x 3 r+y') 2 

 or, on expansion and rearrangement according to powers of r, 

 in virtue of (4), 

 (2v'x 1 *)r 3 + (6u'v'x 1 2 -x 2 2 -3x 3 2 )r*+ 2[x 1 v'(u' 2 -v' 2 ) 



+ 2u' 2 v'x 1 -x&'-3x 3 y']r=Q (5) 

 To make the coefficient of r vanish we must have 



say, x^Zu'W-v'^-x'^IZy' (6) 



Equation (5) is now satisfied by taking 



r= (a;, 1 + 3ay - SuVa;, s )/2n V 

 _ -W + (3u'-v' - vy-xS - 2x'(3u"v'-v'*)x l x t +tf i x t ' > 



(7) 



on substituting the value of x 3 given by (6). 



1 See the writer's paper, Part I. 



