CHAP. II] ONE LINEAR EQUATION IN THREE UNKNOWNS. 71 



G. B. Mathews 142a proved that, if \l/(ri) is the number of positive integral 

 solutions of x + y = n in which 3x ^ 4y, 2x ^ 7y, then 



For the problem in n instead of two unknowns, see Ch. III. 



ONE LINEAR EQUATION IN THREE UNKNOWNS. 



T. F. de Lagny 10 (p. 595) treated py = ax + z by giving values to z 

 which are the successive multiples of the g.c.d. of p and a. The methods of 

 de Paoli 62 and Mac Mahon 48 were given above. 



Several 143 found the 12 sets of positive integral solutions of 



Wx + lly + 12z = 200. 



L. Euler 144 treated Aa + Bb + Cc = 0. For example, 



49a + 596 + 75c = 0. 



Divide by 49 and set a + b + c = d. Thus 106 + 26c + 49d = 0. Di- 

 vide by 10 and proceed as before. We ultimately get all integral solutions: 



a = - Se - 7f, b = 13e + 2/, c = 3/ - 5e. 

 P. Paoli 145 solved 5x -\- 8y + 7z = 50 by successive substitutions : 



x + y = t, 5t + Zy + 7z = 50, 



</ + * = -*', 3f + 2t + 7z = 50, 



t + t' = t", 2t" + *' + 7s = 50. 



Since a coefficient is now unity, the solution is evident. 



A. Cauchy 146 proved that every solution of ax + by + cz = is given by 



x = bw cv, y = cu aw, z av bu, 



if the g.c.d. of a, b, c is unity. 



V. Bouniakowsky 147 proved Cauchy's 146 result by solving 



ax + by + cz = 0, a'x + b'y + c'z = h', a"x + V'y + c"z = h", 

 the two adjoined equations having arbitrary coefficients. Then 



x = bw cv, 



etc., where u = (a'h" - h'a")/A, etc., A being a determinant of order 

 three. _ ___ 



1420 Math. Quest, and Solutions, 6, 1918, 62-64. 



143 The Gentleman's Diary, or Math. Repository, 1743; Davis' ed., London, 1, 1814, 45-7. 



144 Opus, anal., 2, 1785 [1775], 91; Comm. Arith. Coll., II, 99. 



145 Elementi d' Algebra di Pietro Paoli, Pisa, 1, 1794, 162. 



148 Exercices de math., 1, 1826, 234. Oeuvres de Cauchy, (2), 6, 1887, 287. Extr. by J. A. 



Grunert in Archiv Math. Phys., 7, 1846, 305-8. 

 "' Bull. Acad. Sc. St. Pe"tersbourg, 6, 1848, 196-9. 



