152 HISTORY OF THE THEORY OF NUMBERS. [CHAP. Ill 



is (41 12 11 12), since 4, 1 are the coordinates of a referred to the origin A, 

 1, 2, the coordinates of 6 referred to the origin a, and of B referred to the 

 origin c. The nodes in the lower portion Ca- -cDK form a Sylvester 

 regularized graph of the partition (3 2 2 1) ; similarly for the nodes in the 

 upper portion. 



Again, we may think of Sylvester's graph : : , not as representing 



the partition (3 2), but as representing the multipartite number 4, 2. 

 Then consider the partition (42, 31) of the multipartite number 4 + 3, 2 -f- 1. 



By placing the graph of 3, 1 upon the former graph, we obtain a three- 

 dimensional graph of the partition. Such a graph can in general be read 

 in six ways. At the end of the memoir are conjectures as to the generating 

 functions of partitions whose three-dimensional graphs are limited in height, 

 breadth and length. 



R. D. von Sterneck 169 proved Legendre's 23 theorem and deduced from 

 it in a simple way Vahlen's 150 extension. He proved also that, if k is not a 

 triangular number and if we represent k as a sum of integers so that the 

 same part is not used oftener than 3 times in the same representation, then 

 among the representations which contain p distinct parts less often than 3 

 times there are as many sums of even as of odd parts. If %(n h) is not 

 triangular, among the representations of n by distinct summands for which 

 the sum of the absolutely least residues of the summands is = h (mod 3) 

 and in which occur p pairs, each pair being two of three numbers of the 

 form 3m 1, 3m, 3m + 1, there are as many sums of even as of odd parts. 

 Corresponding to the last two theorems there are more complicated ones 

 for triangular numbers. 



J. Franel 170 stated that, if a, b, c are positive integers, relatively prime 

 by twos, and if n is a positive integer, 



(9) ax + by + cz = n 



has n(n + a + 6 + c)/(2ofcc) sets of integral solutions ^ 0, if we neglect a 

 quantity whose absolute value remains, for every n, less than a fixed number. 

 E. Barbette 171 considered (9) for a, b, c positive, a and b relatively prime. 

 If a, j8 are particular solutions of ax + by = 1, then 



x = a(n - - cz) + be, y = /3(n cz) aO 



are the solutions of (9). Let k and h be the quotients obtained when n 

 and c are divided by ab; then the number w of positive integral solutions is 



*[2& - (q + l)/i - 2]g, 



where q is the largest integer ^ n/c. If n is divisible by b, and c is divisible 

 by ab, set H = c/ab and call K the largest integer ^ n/ab; then 



- (g 



189 Sitzungsber. Akad. Wiss. Wien (Math.), 106, Ha, 1897, 115-122. 



170 L'intermScliaire des math., 5, 1898, 54. 



171 Mathesis, (3), 5, 1905, 125-7. 



