INDETERMINATE QUARTIC EQUATIONS 77 



Hence we shall have 



(w 2 +^) 4 =(^ 2 -v 2 ) 4 + (2uv)*+ (x+y}*+ (x-y)*+ (2*/) 4 (1) 

 provided 2uv(u 2 -v*)=x*+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 u=2, 

 i/=l, #=3, y1, and others are easily found, for example, 

 (u, v, x, */)=(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=x l r+u', v=v', x=x 2 r+x', y=x s r+y' (3) 



where we suppose (u' 9 v', x', y') to be a particular solution of 

 (2), i.e. 



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



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



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

 in virtue of (4), 



=Q (5) 

 To make the coefficient of r vanish we must have 



say, ^ 3 =!(3^'V-v /3 )a; 1 - ; r>2|/32/ / (6) 



Equation (5) is now satisfied by taking 



r = 



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



1 See the writer's paper, Part I. 



