160 HISTORY OF THE THEORY OF NUMBERS. [CHAP, in 



R. D. von Sterneck 203 proved De Morgan's 28 result that the number of 

 partitions of n into 3 parts is the integer nearest to n 2 /12 by use of three 

 coordinate axes every pair of which make an angle < 60 and counting the 

 lattice points inside or on the triangle cut out of the plane x + V + z = n 

 by the coordinate planes. Similar use is made of 4-dimensional space to 

 show that the number of partitions of n into 4 parts is the integer nearest 

 (n 3 + 3n 2 - 4)/144. 



N. Agronomov 204 noted that N = 2 a pi l - -pl k is representable as a 

 sum of consecutive integers in (a\ + 1) (a k + 1) ways [Sylvester 119 ]. 



P. A. MacMahon 205 noted that his three-dimensional graphs of plane 

 partitions admit not only of 1, 3 or 6 readings, but may admit just two 

 readings if the weight be == 13. Let each part be =i I and be placed at a 

 node of a two-dimensional lattice with m rows and n columns. The 

 generating function giving as the coefficient of x w the number of partitions 

 of weight w is expressible in six ways, one of them being 



and the other five being derived from this by permuting Z, m, n. A general 

 proof is here first given. The theory of generating functions, especially 

 for I = co , is developed further here and in his next paper. 206 



T. E. Mason 207 proved that 2 a p? 1 - -p a r r , where the p's are distinct odd 

 primes, can be represented as a sum of consecutive integers not necessarily 

 positive in 2(ai + 1) (r + 1) ways. In just one half the representa- 

 tions there is an even number of terms, and in just one half are the terms 

 all positive [Sylvester 119 ]. 



W. J. Greenstreet 208 proved that x + 2y + 3z = Qn has 3ft 2 + 3n + 1 

 integral solutions ^ 0. 



MacMahon 209 showed that the enumeration of partitions of multipartite 

 numbers may be made to depend upon his 141 theory of distributions and 

 symmetric functions of a single system of quantities. 



A. J. Kempner 210 proved that, if 1, Ci, c 2 , form a set of increasing 

 positive integers such that every positive integer is a sum of k or fewer of 

 them, the radius of the circle of convergence of 1 + c\x + c 2 x 2 + is 

 unity. Let every positive integer be a sum of at most k terms of a given 

 set eti, o 2 , ; let cti, |3f be integers such that < on ^= R, /3 | = S, 

 where R and S are any given positive integers; then every positive integer 

 is a sum of fewer than Rl(2kS + k + 1) terms of the set 1, a\a\ + 0i, 

 22 + 02, Finally, the known theorems that any positive integer n 

 is a sum of four squares and that x 2 = 1+3 + 5+ + (2z 1) imply 



203 Rendiconti Circ. Mat. Palermo, 32, 1911, 88-94. 



204 Math. Unterr. 2, 1912, 70-2 (Russian). 



205 Phil. Trans. Roy. Soc. London, 211, A, 1912, 75-110. Memoir V on Partitions. 



206 Ibid., 345-373. Memoir VI on Partitons. 



207 Amer. Math. Monthly, 19, 1912, 46-50. Cf. Sylvester. 119 



208 Ibid., 50-1. 



209 Trans. Cambridge Phil. Soc., 22, 1912, 1-13. 



210 tiber das Waringsche Problem . . . , Diss. Gottingen, 1912. 



