CHAP. XIII] az 2 +fa/ 2 +CZ 2 = 0. 425 



and 



(11) * = XYU 



y A-Y*' 



and conversely. Expressed geometrically, (7) is a cone with the vertex at 

 the origin, and (8) is a plane through the vertex. The intersections are two 

 lines whose projections on the XT/-plane are given by (11). If X , Y , U 

 is a particular solution of (9), and if u , x , y are the values given by (8) 

 and (10), the general solution is 



X = X -x t, Y=Y +y t, U=U -u t. 



Calzolari 123 stated a theorem, which not only decides like Legendre's 114 

 the possibility or impossibility of integral solutions of 



(12) u z = Ax' 2 By 2 (A, B without square factors), 



but determines the general solution without recourse to the process of 

 Lagrange. We may set A = a\-\ ----- \-a 2 m , B = b\-\ ----- \-bl (w^4, n^4). 

 Set Xi = a t x, iji = l>iy. Then 



(13) w 2 



Let pi, - -, p m , #1, -, q n be arbitrary integers. We may set 



Then (13) becomes u '2x i : =F'2yi+K = Q, where 



Ku = (pq} (2z f =t Zj/f) - Sp^f =F Sg#-. 

 In the former give to a;,-, yi their values. Then 



(14) Xi = pi+k, yi = qi+k, u = pq+k. 

 Substitute these values (14) into the two expressions for K. Thus 



(15) (pg) 2 -2p'=F2^ = (mn-l)/c 2 . 



For /fc = 0, values p i} q f satisfying (15) give a;,-, yt from (14) which satisfy (13). 

 Set a = Sai, & = 26i. Then, for k=Q, xa = ^x i ='Zpi p, by q, u = axby. 

 Substitute this u into (12); we get a quadratic for x/y. Hence (12) is 

 solvable if and only if c=Ab 2 Bd 2 =FAB= D, and the general solution is 

 x = abc, y = a? A, u = Abac, where the signs of a, b are ambiguous. 



S. Realis 124 stated that, if A, B, C are relatively prime and without 

 square factors, and if a, |8, 7 give one solution of Ax 2 -\-By 2 +Cz 2 = Q, the 

 general solution is 



where a, b, c are arbitrary. 



w Giornale di Mat., 8, 1870, 28-34. 



124 Nouv. Corresp. Math., 4, 1878, 369-71. 



