CHAP, ill] PARTITIONS. 127 



is (N 2 - l)/8 or (N + 2)(N + 4)/8 according as N is odd or even. An 

 anonymous writer (pp. 93-4) stated that the number of sets of positive 

 integral solutions of x { + + x m = N is {N} m{(N j)/2], where 

 {i} = ('""'"i" 1 ) and j = 1 or 2 according as N is odd or even. 



K. Weihrauch 74 discussed the number f n (A) of sets of solutions of 



... + a n x n = A, 

 where the a's are positive integers. Set 



P = aia 2 -a n , Si = a{+ + a* n , A = pP + m, 

 where m is one of the integers 1, , P. Then 



= p 



f n (A) = f n (m) 



(r-2q)V 



the last being stated without proof, where e is the largest integer ^ r/2, 

 R = m Si/2, and 



= Z -, I ^ (2a +4/3 + 67 + - - = 2s), 



, /s, ...a 



<S 2 r-P2r-l ,p l p 1 p 



" 2rC2r) ! = ' 7 * " w> " ' 



the 5's being Bernouilli numbers. Cf. Meissel 135 and Daniels. 146 

 * E. Meissel 75 treated the partition of very large numbers. 

 E. Lemoine 76 noted that every power n" of an integer n equals a sum of 



n k consecutive terms from 1, 3, 5, 7, -, if /i ^ 2k. Cf. Fre"gier. 22a 



G. B. Marsano's 77 Table 1 is an extension of Euler's table of partitions 



of n into m parts, for n si 103, m ^= 102. Table 2 gives the coefficients as 



far as 53 of the expansions of 



St =' "' ' s " n (1 " XJ] ~ 1 ' 



and the coefficients as far as x 107 of the expansion .of the first ten functions. 

 The results for S/(l x) give the number of ways of partitioning a number 

 into parts 1, 1', 2, 3, -. Those for S/(l - x)(l - z 2 ), into parts 1, 1', 

 2, 2', 3, 4, .... 



74 Untersuchungen Gl. 1 Gr., Diss. Dorpat, 1869, 25-43. Zeitschrift Math. Phys., 20, 1875, 



97, 112, 314; ibid., 22, 1877, 234 (re = 4); 32, 1887, 1-21. 



75 Notiz iiber die Anzahl aller Zerlegungen sehr grosser ganzer positiver Zahlen in Summen 



ganzer positiver Zahlen, Progr., Iserlohn, 1870. 



76 Nouv. Ann. Math., (2), 9, 1870, 368-9; de Montferrier, Jour, de math. e"lem., 1877, 253. 



77 Sulla legge delle derivate generali delle funzioni di funzioni e sulla teoria delle forme di 



partizione de'numeri interi, Geneva, 1870, 281 pp. Described by A. Cayley, Report 

 British Assoc. for 1875 (1876), 322nl; Coll. Math. Papers, IX, 481-3. 



