80 HISTORY OF THE THEORY OF NUMBERS. [CHAP, n 



received the same total amount of money. How many did each sell at 

 first and what were the two prices? The answer given is that at first A 

 sold 7, B 28, C 49 at 7 eggs per kreuzer; the remainder were sold for 3 

 kreuzer per egg. Thus they received 1 + 9, 4 + 6, 7 + 3 kreuzer each. 

 There 191 are eleven sets of positive integral solutions of 



x + y + z = 56, 32x + 20y + IQz = 22-56. 



T. F. deLagny 10 (p. 583) treated the problem of Diophantus 192 II, 18, 

 to find three numbers such that if the first gives to the second 3- of itself + 6, 

 the second gives to the third of itself + 7, the third gives to the first j- 

 of itself + 8, the results after each has given and taken shall be equal. 

 To avoid fractions call the numbers 5x, 6y, 7z. Then the first gives x + 6 

 and receives z + 8 and becomes 4x + z + 2. Thus 



2 = 5y + x -1 = 6z + y -1. 



Eliminating z and y in turn, we get 



19s + 18 17x + 12 



y 



26 ' ~ 26 



Their difference (2x + 6)/26 must be an integer. Multiply it by 8 and 

 subtract from z; thus x 36 and hence x 10 is divisible by 26. Since 

 2x + 6 and 2(x 10) are divisible by 26, while their difference is 26, the 

 problem is possible. We may take x = 10 + 26& and get an infinity of 

 integral solutions. He employed the same method to treat any such 

 " double equalities " of the first degree, which may be reduced to 



ax =fc q bx zb d 



y = - -, z = - 



P P 



The principle is to get x c by elimination. 



N. Saunderson 13 (pp. 337-354) and A. Thacker 193 treated two equations 

 in x, y, z in the usual way. 



L. Euler 194 discussed the regula Coeci. Given 



p + q + r = 30, 3p + 2q + r = 50, 



eliminate r. Thus 2p + q = 20, whence p may have any value ^ 10. 

 In general, for 



(1) x + y + z = a, fx + gy + kz = 6, / ^ g ^ h, 



b ^ f(x + y + z} = fa, b ^ h(x + y + z) = ha, 



while b must not be too near these limits fa, ha. By eliminating z, we get 

 ax + (3y = c, where a and 13 are positive. A similar pair of equations in 



191 Ladies' Diary, 1709-10, Quest. 8; C. Button's Diarian Miscellany, 1, 1775, 52-3; T. 



Leybourn's Math. Quest. L. D., 1, 1817, 5. 



192 Diophantus used 5x, 6x, 7x and got x = 18/7. G. Wertheim, in his edition of Diophantus, 



1890, proceeded as had de Lagny. 



193 A Miscellany of Math. Problems, Birmingham, 1, 1743, 161-9. 



194 Algebra, II, 1770, Cap. 2, 24-30; 1774, pp. 30-41; Opera omnia, ser. 1, 1, 1911, 339-344. 



