72 HlSTOKY OF THE THEORY OF NUMBERS. [CHAP. II 



V. A. Lebesgue 148 noted that, if the g.c.d. of a, b, c is unity, all solutions of 



(1) ax + by + cz = d 



are given by 



x = da8 + can + vb/D, y = d{38 + c@u va/D, z = dy Dw, 



where ^ and y are arbitrary, aa -\-b^ = D, D being the g.c.d. of a, 6, and 



D8 + CT = 1. 



H. J. S. Smith 149 stated that if a, 6, c and a', b', c' are two sets of solutions 

 of Ax + By + Cz = 0, where A, B, C have no common divisor, the com- 

 plete solution is 



x = at + a'u, y = bt + 6'w, 2 = ct + c'u, 

 if and only if there is no common divisor of 



be' b'c, ca' ac', ab' a'b. 



A. D. Wheeler 150 treated (1) by taking 1, 2, for z until we reach a 

 value for which axi -\-byi d cz < a + b and hence is not solvable. 

 By simplifying this method, he found the number of solutions. 



L. H. Bie 151 expressed the general solution of (1) in terms of the residues 

 of d pc modulo b. 



C. de Comberousse 152 employed the g.c.d. 8 of a, b. Let the g.c.d. of 8 

 and c divide d. Then d cz = 56 has an infinitude of solutions z, 6. For 

 each d, xa/d + yb/d = 6 has an infinitude of solutions. If a, j8, y is one set 

 of solutions of (1), every solution is given by 



x = a - bd + c6 r , y = /? + ad + c6", z = 7 - aB' - be" (6, 6', 6" arbitrary). 



A. Pleskot 153 treated (1) by continued fractions. 



While in various books 154 on algebra the solution of (1) involves three 

 parameters, that by G. M. Testi 155 involves only two. Let the greatest 

 common divisor d of a and b be prime to c. Then 



a b 



T^ + ~2/ = ^, 8t + cz = d. 



The second has the general solution to c0, Z Q -f- 50, if to, z is one solution. 

 All solutions of the first are given by x = x t 6b/8, y = y t + da/8, if x , 

 y is a solution of 



a b 



148 Exercices d'analyse num6rique, Paris, 1859, 60. 



149 British Assoc. Report, 1860, II, 6; Coll. Math. Papers, I, 365-6. 

 ""The Math. Monthly (ed., Runkle), New York, 2, 1860, 407-410. 



161 Tidsskrift for Mat., 2, 1878, 168-78. 



162 Algebre superieure, 1, 1887, 179-183. 



163 Casopis, Prag, 22, 1893, 71. 



154 Cf. J. Bertrand, Trait6 616m. d'algebre, 1850; transl. by E. Betti, Florence, 1862, 285. 

 Periodico di Mat., 13, 1898, 177. 



