126 HISTORY OP THE THEORY OF NUMBERS. [CHAP, in 



A. Cayley, 67 denoting by P t the number of partitions of n into i parts, 

 proved that 



1 -P 2 + 1-2P 3 - (n - l)!P n = 0. 



For, the number of partitions n = aa + 6/3 + is 



n! 



Multiply this by ( l) p ~ 1 (p 1)! and sum for the sets of solutions of 

 p = a -\- -f- . . . ; we get the initial theorem. 



A. Vachette 68 stated that one of n 2 , n 2 1, n z 4, n 2 + 3 is divisible 

 by 12 and the quotient is the number of sets of integral solutions > of 

 x + y + 2 = n [De Morgan 28 ]]. 



L. Bignon 69 noted that the respective cases occur for n = Qn', Gn' + 1 

 or 5, 6n r -f 2 or 4, 6n' + 3. For n = Qn', for example, he separated the 

 sets of solutions into n/3 sets each with y x a constant 0,1, , \n 2, 

 and exhibited the solutions of each set. 



E. Catalan 70 noted that x\ + + x n = s has (^ij) sets of positive 

 integral solutions. Subtract unity from each x and apply his 25 former 

 result. 



Let 71 (n, g) be the number of partitions of n into q distinct parts, [n, q~\ 

 into q equal or distinct parts. Proof is given of theorems of Euler: 



(n, g) = (n - q, q - 1) + (n - q, g), (n, q) = [^ - gg 2 , g J , 



[n, g] = Z [> - Q, ij (w, ff) = Z (^ ~ i?i ? ~ 1), P = \ 1 1 



t=i f=i L 5 



and the first written for [ ]. Here n ^ 2g. 



In Xi + + x q = n, Xi ^ x z = - = x q , take Xi = a ^ 

 set ^ = I/,- + a 1 (i = 2, -, g). Then 



2/2 + + 2/a = n - 1 - (a - 

 for y's > 0. Hence 72 



[n, g] = Z [^ - 1 - (a - l)g, g - 1], a = [n/g]. 



a=l 



Taking q = 3, he deduced the result of De Morgan 28 and Vachette. 68 



C. Hermite 73 stated that the number of sets of positive integral solutions 

 of 



x + y + z = N, x ^y + z, y ^ z + x, z ^x + y 



67 Math. Quest. Educ. Times, 7, 1867, 87-8; Coll. Math. Papers, VII, 576-8. 



68 Nouv. Ann. Math., (2), 6, 1867, 478. 

 Ibid., (2), 8, 1869, 415-7. 



Melanges Math., 1868, 16; M6m. Soc. Roy. Sc. Liege, (2), 12, 1885, No. 2, 19. 



71 Ibid., 62-65; M6m. Liege, 56-58. 



72 Ibid., 305-12; M6m. Liege, 264-71. Nouv. Ann. Math., (2), 8, 1869, 407. 



73 Nouv. Ann. Math., (2), 7, 1868, 335. Solution by V. Schlegel, (2), 8, 1869, 91-3. 



