70 HISTORY OF THE THEORY OF NUMBERS. 



t 



and, for m odd, 



[CHAP. II 



a 



a 



[ r (m-l) ~| 

 TJ+- 



Then x = x r + (- 



, y = y' + (- l} m+l arj, -f 17 = w (m >/(a6). 



L. Crocchi 137 noted that, if in Hermite's 119 process we have reached the 

 dividend n ( *> = aQ + r p = 6Q' + r' P , then n (p+1 > = n (p) - r p - TV. For 

 example, consider 5x + lly = 488. 



Residues 



Quotients 



Here 481 = 488-3-4, etc. Thus 



x' = 97 - 96 + 94 - 92 + 90 - 90 + 88 = 91, 



y' = 1 + 1 + 1 + 40 = 43, x = 91 - llm, 



y = 43 - 5n, m + n = 440/(5-ll) = 8. 



To find x' more readily, use the second, fourth and sixth entries 8, 9, 10 

 in the third column and set 



'+[!]-* '-[?]-* 



7 4 = 1 + [|] = 2, x' = 97 - / 2 - J 4 - J. = 91. 



Similarly, from the second column, y' = 44 1 = 43. But if the 

 number of operations had been even, we would have used /i, /a, I&. 



L. Rassicod, 138 V. A. Lebesgue, 139 G. Chrystal, 140 L. Aubry 141 and E. 

 Cesaro 142 evaluated co by known methods. Cf. Laguerre 91 of Ch. III. 



m II Pitagora, Palermo, 15, 1908-9, 29-33. 

 188 Nouv. Ann. Math., 17, 1858, 126-7. 



139 Exercices d'analyse numdrique, 1859, 52-3. 



140 Algebra, 2, 1889, 445-9; ed. 2, vol. 2, 1900, 473-6. 



141 L'enseignement math., 9, 1907, 302. 



142 M6m. Soc. Roy. Sc. de Liege, (3), 9, 1912, No. 13. 



