412 HISTORY OF THE THEORY OF NUMBERS. [CHAP, xui 



* Ferval 70 gave an infinitude of solutions of each of the equations 



A. Hurwitz 7 ' called rfs and u/v a pair of approximating fractions for a 

 number between them if us vr = l. If 0<??z<2VZ) and if at least one of 

 A, C is positive, and D = B- AC>0, every pair of integral solutions of 

 Au 2 +2Buv-)rCv 2 = m is such that u/v is an approximating fraction to one 

 of the roots of Ax z +2Bx+C = 0. If both A and C are negative, we get 

 the same result by assuming also that v 2 > A/(2 VD m). 



H. Scheffler 72 made successive additions to get p, 2-p, &p, and then 

 a table of values for pn z +p^n\. The aim is to solve ax 2 -\-bxy -\-cy* = q. 



R. W. D. Christie 73 solved x z +xy y z = 1 by use of continued fractions. 

 Cf. J. Wasteels 72 of Vol. I, p. 405, of this History. 



A. Cunningham and Christie 74 solved y z avy av 2 = l. 



A. Levy 75 recalled the special case of Dirichlet's theorem on the units 

 of an algebraic field, that if (a, b) is the least positive solution +(1,0) of 

 ky 2 = 1, where k is a positive integer, every solution (u, v) is given by 



) n , co 2 co k = 0. 



Several writers 76 solved x z +xy-\-y- = l. 



C. Ruggeri 77 used the series with the recursion formula z n+1 = z n -\-z n -i 

 to solve ax 2 bxy-\-cy 2 = k, when 6 2 4ac = 5m 2 . 

 See papers 88, 89; also Leslie 90 of Ch. XII. 



SOLUTION OF Ax' 2 +2Bxy+Cy*+2Dx+2Ey+F = Q. 



L. Euler 78 noted that if Ax 2 +2Bxy+Cy"+2Dx+2Ey+F = Q has the set 

 of solutions x = a,y = b, and if A = B 2 A C > 0, so that p 1 Ag 2 + 1 is solvable, 

 a second set of solutions is 



x = a(p+Bq) + bCq+Eq+(p-l)(BE-CD')/A, 

 y = b(p-Bq}-aAq-Dq+(p-l)(BD-AE)/&. 



J. L. Lagrange 79 showed how to find the rational and integral solutions of 

 (1) ax z +(3xy+ 7 y z +dx+ey+{ = 0. 



Solving it algebraically for x in terms of y, we get 



70 Jour, de math, spec., 1889, 94, 141. 



71 Math. Annalen, 44, 1894, 425-7. 



72 Vermischte Math. SchriJften, Part II, Die Quadratische Zerfallung der Zahlen durch 



Differenzreihen, Braunschweig, 1897, 28-59. 



73 Math. Quest. Educ. Times, 73, 1900, 71. 



74 Ibid., (2), 10, 1906, 24-25. 



75 Bull, de math. 61e"m., 15, 1909-10, 113-5. Cf. J. Sommer, Vorlesungen iiber Zahlentheorie, 



1907, 100-7; French transl. by L6vy, 1911, 103-113. 

 7 Amer. Math. Monthly, 15, 1908, 44. 



77 Periodico di Mat., 25, 1910, 266-276. 



78 Novi Comm. Acad. Petrop., 11^1765 (1759), 28; Comm. Arith., I, 317; Op. Orn., (1), III, 76. 

 Mem. Acad. Berlin, 23, anne6 1767, 1769, 272; Oeuvres, II, 377-381, 509-522. Cf. his 



simplifications in his additions to Euler's Algebra, 2, 1774, 554, 595-607; Oeuvres de 

 Lagrange, VII, 113, 140-7; Euler's Opera Omnia, (1), I, 593, 615-22. Cf. Smith. 88 



