1876.] Verification in the Partition of Numbers. 253 



Y. Z(x—2xy + 3xijz—&q.) = \fm, 



where xpn denotes, as also in VI. and VII., the number of divisors of ??, 

 unity and n included. This formula does not verify the value of N, 

 but it verifies S#, 2#y, &c. ; for n=9, \pn=3 ; and it gives 



67-2.47+3.10 = 3. 



VI. 2(l-2y + 3yz-4yziv + &c.)=\P(n + l)—^n. 



Tor «=9 (2ty, the total number of 2's) =26, 2yz==9, 2yzw==l, and 

 30-2.26 + 3.9-4.1=4-3. 



VII. 2(l-2[ 2 /] 1 + 3M 2 -4[^] 3 + &c.)=^+2)-l, 



where [ ] r denotes that the enclosed quantity is zero for every partition 

 which does not contain at least r l's. 

 For *i=9, SCyl-19, %4=^ and 



30-2.19+3.3=2-1. 



Vin. N-l = 2(^-l-^-2.7/+o7-3.^-&c), 



where, as throughout, the point is written in place of brackets (ex. gr. 

 x — 2.y stands for (x—2)y) i and a negative factor is to be treated as 

 zero. 



Tor n=9, 2(>-l)=45, 2(a?— 2.y)= 17, 2(>-3 .?/z) = l, and 



30-1=45-17 + 1. 



IX. N-w=2({^-3} 2 + {^-5} 3 + {*-7} 1 + &c.), 



where { } r denotes that the enclosed quantity is zero for every partition 

 in which an element >r appears. 



Eor7i = 9,S{^-3} 2 = 2 + 4 + 6 = 12,S{ t r-5} 3 =l + 2 + 4 = 7 3 2{.r-7} 4 

 as 2 ; and the formula gives 



30-9 = 12 + 7 + 2. 



X. N-n=2({«-l.y} a +{fl?-3.j^} 8 +{*-6.yaw};+&c., 



where { } r has the same meaning as in IX., and 1, 3, 6 ... . are the tri- 

 angular numbers. 



For n = 9, the formula gives 



30-9=20 + 1. 



XL If wbe uneven, 2(l-[^ + l] 1 -[^+l. 2/ + l.^+l] 3 -&c.)=0; 

 and if n be even, s(l -[l]°-[> + l .y+l] 2 _[^+l . . . w;+l] 4 -&c.) = 0, 



where a? = the number of 2's, y = the number of 4's &c. in any parti- 

 tion, and [] r denotes that the enclosed quantity is equal to zero unless 

 the partition contains exactly r ones, and no other uneven element. 

 Thus 2[1]° is equal to the number of partitions formed wholly of even 

 elements. (#, y, z . . . may, of course, have zero values.) 



