286 ProE. E. T. Bell on Parametric Solutions for a 

 arbitrary functions 



&=««) = 1,2,3,4), .... (2) 



where u is a parameter. In (1) the a {j are functions o£ 

 the Xi. 



2. For brevity we shall assume that the right-hand 

 member of (1) is reducible, in the usual way, to a sum of 

 four squares, 



4 



(U 2 = £ ( hn dx l + b i2 ch\ 2 + b i3 dx s + b i4: dx^f, . (3) 



subject to the condition that the determinant of the system 

 budx l + b i2 dx 2 + b is dx, + b {i dx A = </>,- (u) (i = 1, 2, 3, 4), . (I) 



in which dx^ dx 2i dx%, dx± are the unknowns, does not 

 vanish. If (1) is such that this reduction is impossible ; 

 viz., if the aij are such that no by exist such that the deter- 

 minant of (4) is different from zero, a slight and obvious 

 modification of the algebra leads to a parametric solution of 

 essentially the same sort as that now given. 



3. The determinant of (4) not vanishing, we can solve for 

 the dxi, getting 



dZi^Bnfafa) +Btofa(u) + B & <l> z (u) +B^(w) (t = l, 2, 3, 4), 



... (5) 



in which By are functions of the &#, hence of the a^ and 

 therefore of the Xi alone. Hence, integrating, 



^•^(Bfl^^ + Ba^^ + Bis^sW + Btt^WyM ; • (6) 



or when the J$i- 3 are independent of u, 



4 



^SB^(«)iu (i=l, 2,3,4). . . (7) 



We shall now determine 5 and the </>/(V) in terms of the 

 functions (2) so that on substituting in (6), 5 and the 

 resulting xi reduce (1) to an identity. There are many 

 ways of doing this ; one of the most obvious is 



*=J(?i 2 +& 2 + ft 2 +?/)*', 



