1884.] Mr Glaisher, On the sum of the divisors of a number. 119 



E{n) 



-2E(n-l) + 2E(n-2) 



- 3E (n - 3) + 3E(n - 4) - 3E(n - 5) 



-4E(n-6) + 



...:;:;;;;::;;;;::;;;:;;;;:;;:;;;::;;;;;;:+ , (^p^' ( ;i)'"" 



= or (-lr 1 ^, 

 according as s is uneven or even, s having the same meaning as in 

 §§ 5 and 9. 



§ 12. Corresponding to the ^-formula in § 8 we have the 

 formula 



E(n)-E(n - 1) + E (n - 3) - E(n - 6) +E(n - 10) - ... 



= or (- If x I {(- l)i{V(8»+D -i} x ^( 8to + 1) - 1}, 



according as n is not, or is, a triangular number*. 



§ 13. The three corresponding formulge of §§ 5, 9 and 11 may- 

 be enunciated in a uniform manner as follows : 

 If n be any number, then 



(i) 



cr{n) 



- 2a (n - 1) - 2a (n - 2) 



+ 3a - 3) + 3o- (n - 4) + 3a (n - 5) 



-4o-(?i-6) 



+ (-r 1 scr(0) =0, 



where <r(0) denotes ^(s 2 — 1). 



(ii) 



?(») 



+ 2£{n-l) + 2£(n-2) 



+ 3t;{n - 3) + 3£ (n - 4) + 3£(rc - 5) 

 + 4£(™-6) + 



+sr(o)=o, 



where a- (0) denotes — £ (s 2 — 1). 



* The formulae in §§ 11 and 12 are proved in a paper on the function E (n) 

 communicated to the London Mathematical Society on February 14, 1884. 



