418 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xm 



To solve (pp. 127-221) F+2dX+2eY+f=0, where F=aX*+2bXY+cY 2 , 

 A = 6 2 ac^O, it is transformed as usual into F = M, which is treated as 

 usual by the theory of binary quadratic forms. If A = 0, it is transformed 

 into u z +r = Py, which is of the type first treated. In each case there is a 

 discussion as to which qf the solutions are integral. 



H. J. S. Smith 88 noted that Euler's 81 - 82 methods are incomplete for the 

 reasons noted in Ch. XII, Smith. 139 He modified Gauss' 85 method by em- 

 ploying the g.c.d. 5 of a, j3, 7, and employing the new variables X=p/5, 

 Y = q/d. Thus aX 2 +2bXY+cY 2 = a'& f , where a' = a/8, A' = A/5. Then if 

 X n , Y n is any representation of a'A' by (a, b, c), we separate the integral 

 from the fractional solutions x, y by separating (by Lagrange's method) 

 those values of X n , Y n which satisfy the congruences X n 0/5 = 0, 

 Y n 7/5=0 (mod a') from those which do not, and obtain a finite number 

 of formulas exhibiting all integral solutions. 



G. Wertheim 89 treated (1) as had Lagrange, 79 and by reducing it to 

 ax z +2bxy+cy 2 = M and then applying the theory of binary quadratic 

 forms. 



C. de Comberousse 90 treated (1) for the case 7 = 0, whence y = Q/L, 

 where Q is a quadratic and L a linear function of x. Thus L must divide 

 a certain constant N, whence set L = d, d any divisor of N. 



Rautenberg 91 reduced the solution of an equation of degree two in 

 two variables to Bx*-\-Cx-\-D= D and gave other known results. 



R. Marcolongo, 92 G. B. Mathews, 93 P. Bachmann, 94 and E. Cahen 95 

 treated (1). 



Focke 96 gave the usual application of quadratic forms to our problem. 



E. de Jonquieres 97 showed by detailed examples that the methods of 

 Lagrange (continued fractions) and Gauss (period of reduced forms) for 

 solving indeterminate equations of the second degree are less different 

 than they seem, since they employ the same auxiliary quantities, and rest 

 on the development of practically the same ideas. 



G. Bisconcini 98 noted that x = y = 2 is the only positive integral solution 

 of xy = x-}-y, and re = 0, 1, 2/ = 0, l,the only integral solutions of x 2 -\-y 2 = x+y. 



J. Westlund" proved that x*+y z = (2x 1)/3 is impossible in integers. 



C. Ciamberlini 100 stated that (x+y)(x+y-}-l)+2y = a has a single posi- 

 tive integral solution if a is a positive integer. 



T. Pepin 101 used the method of Gauss. 85 



88 British Assoc. Report, 1861, 313; Coll. Math. Papers, 1, 1894, 200-2. 



89 Elemente der Zahlentheorie, 1887, 226-236, 369-374. 



90 Algebre sup&ieure, 1, 1887, 185-191. 



91 Ueber dioph. Gl. 2 Gr., Progr. K. Gymn., Marienburg, 1SS7. 

 92 Giornale di Mat., 25, 1887, 161; 26, 1888, 65. 



93 Theory of Numbers, 1892, 257-261. 



94 Arith. der Quad. Formen, 1898, 224-231. 



95 Elem. de la th6orie des nombres, 1900, 286-299. 



96 tiber die Auflosung d. dioph. Gleich. mit Hilfe der Zahlentheorie, Progr. Magdeburg, 1895. 



97 Comptea Rendus Paris, 127, 1898, 694-700. 



98 Periodico di Mat., 22, 1907, 121-2. 



99 Amer. Math. Monthly, 14, 1907, 61. 

 100 Suppl. al Periodico di Mat., 11, 1908, 104-5. 



101 Mem. Pont. Accad. Nuovi Lincei, 29, 1911, 319-327. 



