CHAP, ill] PARTITIONS. 135 



a, , L The denumerant thus equals 2\=i Wi, where the wave Wi is 

 the residue of 



2r~ n e nx F(r q e- x ) = 



summed for the primitive 5rth roots r q of unity (or for their reciprocals). 

 Now make the important substitution v = n + (a + + Z)/2. Then 



Wi = residue of 2r v q e vx /IL(r a g l2 e a * 2 - r g - (a/ V (aa:/2) ) , 



the product extended over the similar terms in a, b, , I. Expanding the 

 summands into power series, we see that each wave and hence the denumer- 

 ant is a sum of products of polynomials in v each multiplied by a quantity 

 c2(r v+s r"~ 5 ), where 5 is one-half of the number $({) of integers < i 

 and prime to i (since Wi becomes Wi when v is changed in sign) . Give 

 to each such term of the denumerant an undetermined coefficient, as 



+ B + (- iyC + D2(r v+1 + r"- 1 ), r 2 + r + 1 = 0. 



1, 2, 3, 



Write s = d + + Z (s = 6 in this case). It is shown that the denumer- 

 ant is zero for all values of v from to f s 1 inclusive if s be even, and for 

 all values from f to f s 1 inclusive if s be odd. This fact serves to deter- 

 mine uniquely the ratios of undetermined coefficients. For example, in 

 the above case, v = 0, 1, 2, and 



B + C - 2D = 0, A+B-C + D = Q, 4A+B + C + D = 0, 



whence A = 60-, B = - 7<r, C = - 9<r, D = - 8<r. The value 9A + B 

 C 2D for v = 3 must be unity. Hence o- = 1/72. Since v = n + 3, 

 the result agrees with that given by De Morgan. 28 The case of the elements 

 1, 2, 3, 4 is treated similarly. The wave Wi is discussed in detail. Applica- 

 tion is made to the number of sets of solutions of 



where a\, , a t - are relatively prime in pah's. For i = 2, the number is 

 (did-i ai a 2 1)12. 



Sylvester 108 noted that there is a one to one correspondence between 

 the indefinite partitions of n with parts in ascending order and the series 

 0, , n such that each term is not greater than the mean between its 

 antecedent and consequent. 



If d and 6 are incommensurable, integers x, y can be found such that 

 ax + by + c is indefinitely small. If it be impossible to find integers 

 X, n, v such that 



X(&7 Co) + n(ca 07) + v(dy ba) = 0, 



ax + by + cz + d and ax + py + 72 + 5 may simultaneously be made 

 arbitrarily small by choice of integers x, y, z. Cf. Jacobi 256 of Ch. II. 



108 Johns Hopkins Univ. Circ., 1, 1882, 179-180; Coll. Math. Papers, III, 634-9. First theorem 

 also in Math. Quest. Educ. Times, 37, 1882, 101-2. 



