CHAP, xiii] QUADRATIC EQUATION IN n^3 UNKNOWNS. 431 



C = ab-h~, 

 F = gh-af, 

 G=fh-bg, 



where q is arbitrary. Multiply these by , 77, f and add. Thus 



(abcfgh) fa ft (xyz) = (dbcfgh) (toff, 



so that we get (4). Using the multipliers C, F, G, we get z = f. Then the 

 first two equations readily give 



which satisfy (4) identically with z' = z = l. Taking /i = 0, we get values 

 making ax 2 +2gx+c-{-by~-{-2fy identically equal to the same function of 

 x r , y'. To pass to formulas exactly equivalent to Euler's, set 



(l-q 2 ab)/(l+q 2 ab)=s = ^l 



H. J. S. Smith 153 stated criteria for the solvability of (1) in integers, 

 whether the coefficients are real integers or complex integers p-\-qi. It 

 suffices to consider the case in which the coefficients a, , 6 2 of / have no 

 common divisor, while / is an indefinite form of determinant 4= 0. Let ft be 

 the g.c.d. of the nine two-rowed minors of the determinant ft 2 A of /. Let 

 SlF be the contra variant (6 2 ai 2 )^ 2 + of /. Let ft, A, ftA be the quo- 

 tients of ft, A, ftA by the greatest squares contained in them respectively. 

 Let co be any odd prime dividing ft, but not A ; 5 one dividing A but not 

 ft; 6 one dividing both ft, A. Then /=0 is solvable in integers =1=0 if and 

 only if 



(?HD. 



where the symbols in the left members are those of Legendre, and those 

 in the right members are generic characters of / (Eisenstein, Jour, fur 

 Math., 35, p. 125). This theorem is a generalization of the criteria for the 

 solvability of ax z -\-aixl-{-azxl = 0. 



A. Meyer 154 proved the preceding theorem for forms of odd determinant. 



P. Bachmann 155 proved that, if F is a ternary quadratic form, all solu- 

 tions of p 2 F(q, q', q")=2 h 5, in which p is divisible by 5, are obtained 

 by a definite rule from any one solution and all solutions of 



The left member repeats under multiplication. 



S. Realis 156 stated that all integral solutions of x 2 -\-ny z = u--\-nv 2 are 

 given by (cf . Gerardin 167 ) 



y= (7 a) 2 



163 Proc. Roy. Soc. London, 13, 1864, 110-1; Coll. Math. Papers, 1, 1894, 410-1. 



164 Jour, fur Math., 98, 1885, 177-9. 



165 Jour, fiir Math., 71, 1870, 299-303. 



156 Nouv. Ann. Math., (2), 18, 1879, 508. 



