CHAP, xiii] QUADRATIC EQUATION IN n^3 UNKNOWNS. 433 



and (II) hold and either abcd=2, 3, 5, 6, 7 (mod 8); or abcd=l and 

 a _|_5_j_ c _[_d = (mod 8); or abcd=4: (mod 8) and, if a and b are even and 

 c and d odd, either labcd=3, 5, 7 (mod 8), or \abcd =1 (mod 8) and 



a .b , (cd) z -l 



- -- (mod 8). 



He gave necessary and sufficient conditions for integral solutions of / = 0, 

 where / is any quaternary quadratic form. Finally, 



ax z +by z +cz z +du z +ev z = Q 



is solvable in integers not all zero if the coefficients are odd and not all of 

 the same sign. 



H. Minkowski 161 defined an invariant D in terms of the prime factors 

 of the determinant of the quadratic form and proved that zero can be 

 represented rationally by every indefinite quadratic form in 5 or more 

 variables, by one in 4 variables if D is not divisible by the square of a prime, 

 by one in 3 variables if D = l, and by one in two variables if D= 1. 



G. de Longchamps 162 would solve z 2 2X;=2X;?/- by choosing integers 

 x, i, , a n for which 2\ i a 2 i = 2x'2\iai (for example, by taking a i} , 

 o>n-2 arbitrary even integers and choosing a n -\ so that ZSJ^Aien 2 is divis- 

 ible by X B , and taking a n to be the quotient) and then finding y i} , y n 

 from xyi = <xi. Application is made to nx z = y z -\-(n l)z z and to 



x 2 xy+y z =z z . 



The discriminant of the latter equation in x is y z 4Q/ 2 z z ), which must 

 be a square /c 2 ; whence a solution is z = 7, y = 5, A; = 11, x = S. 



P. Bachmann 163 proved Meyer's 160 theorems. 



A. Meyer 164 discussed the solution of p z ttF(q, q', q") = e [cf. Bach- 

 mann 155 ]. For this and the next paper, see the chapter on quadratic forms. 



G. Humbert 165 treated the integral solutions of x 2 4yz 4tu = A. 



Anonymous writers 166 stated that x z +y z z z = 2u z has the solutions 



x = 2ak(c z -ak) } y = c z (c z -4ab'), z= {c 2 -2a(a+6)} 2 -2a 2 (a 2 -26 2 ), 



Or we may compare the known solutions of y z z* = h 2 , x 2 +h z = 2u z and 

 take h = a?b 2 = 2m 2 l 2 ; hence an infinitude of solutions can be found 

 from one. 



A. Gerardin 167 stated that x*+hy*=z*+kt z has the solutions (Realis 156 ) 



= n z +hp z hm z , y = 



m)j t = hp z +hm z n z +2p(n 



161 Jour, fiir Math., 106, 1890, 14. Gesamm. Abhandl., I, 227. 



162 El Progreso Mat., 4, 1894, 40-7; Jour, de math, elera., 18, 1894, 5. 



163 Arith. der Quad. Formen, 1898, 259-266, 553. 



164 Jour, fur Math., 116, 1896, 321. 



165 Jour, de Math., (5), 9, 1903, 43. 



166 Sphinx-Oedipe, 1907-8, 30, 95-6. 



167 Ibid., 107-9. 

 29 



