134 HISTORY OF THE THEORY OF NUMBERS. [CHAP, ill 



parts, the first one being {3} and without three consecutive numbers at the 

 end. We may suppress these partitions. In general, consider a partition 

 commencing with {n} and ending with n consecutive numbers. If the 

 first term is n, efface it and add 1 to each of the last n numbers, which can 

 be done unless the number of parts is ^ n, whence w = n(3n l)/2. If 

 the first term is n + 1 and if the last n + 1 terms are not consecutive, 

 reduce by 1 each of the last n and place n before the first part, which can 

 be done unless the number of parts is n, whence w = n(3n + l)/2. Hence 

 E = unless w n(3n l)/2, and in that case E = 1, there remaining 

 a single partition into n parts. For an exposition of this proof, with illus- 

 trative graphs, see E. Netto, Lehrbuch der Combinatorik, 1901, 165-7. 



A. Capelli 106 considered a matrix (a*,) of n 2 integers ^ such that the 

 sum of the numbers in each row or column is always m: 



+ (X iZ + + din = Cfij + 2 y + ' + <Xnj = m. 



The number of these matrices equals the number of linearly independent 

 forms derived from the general form in n sets of variables, homogeneous 

 and of degree m in each set of variables, by means of the operation 2 77 t -d/d,-, 

 where the and ?/ are two of the n sets. 



Several 1060 found the number of ways 34 is a sum of four distinct positive 

 integers. 



J. J. Sylvester 107 gave an exposition of the theory previously only 

 sketched by him. 43 Employing Cauchy's term residue to denote the coeffi- 

 cient of I/a; in the expansion of a function of x in ascending powers of x, he 

 considered any proper rational function F(x), so that the degree of the 

 numerator is less than that of the denominator. Then we may write 







x 



The residue of *L v Jl\F(a v e*) is easily seen to be the constant term of F(x). 

 Hence if x~ n f(x) is a proper rational function, the coefficient of x n in the 

 rational function f(x) is the residue of 2r~ n e nx f(re~ x ), summed for each 

 value r 4= of x making f(x) infinite [as the a's for F(x)~], The " denumer- 

 ant to the equation ax + + It = n" denoted by 



n 



a,b, -,1,' 



is the number of sets of integral solutions ^ of the equation, and equals 

 the coefficient of x n in the expansion of 



F(x) = (1 - x a )~ l ---(l - x 1 )- 1 . 

 Let 5i = 1, 5 >, , S M be the integers dividing one or more of the numbers 



104 Giornale di Mat., 19, 1881, 87-115. 



106a Math. Quest. Educ. Times, 34, 1881, 51. 



107 Amer. Jour. Math., 5, 1882, 119-136 (Excursus on rational fractions and partitions). 



Johns Hopkins Univ. Circ., 2, 1883, 22 (for the first theorem). Coll. Math. Papers, 



III, 605-622; 658-660. 



