1884.] Mr Glaisher, On the sum of the divisors of a number. 113 



«r(6) 



-2M5) + <r(4)} 



+ 3{c7(3) + -(2)+c7(l)} = (-l) 4 K4 3 -4) ) 



«r(7) 



-2{o-(6) + <r(5)} 



+ 3 {o- (4) + o- (3) + a- (2)} 



-Ml) =(-l) 5 H4 3 -4) 



that is, 6- 2 { 7 + 4} +3 {3 + 1} =-4, 



12-2{ 6 +7} +.3 {4+ 3 + 1} =10, 



8 - 2 (12 + 6} + 3 {7 + 4 + 3} - 4 = 10. 



The expression ^ (s 3 — s) is equal to the sum of the first 5—1 

 triangular numbers so that the series is always numerically equal 

 to the sum of the triangular numbers which do not exceed ??. 



If the series were continued one term further the next term 

 would be (— l) s " 1 scr(0), and if we put cr(0) = i (s 2 — 1) this term 

 becomes (- l) s_1 £ (s 3 - 5). We may therefore, by employing a 

 convention of the same kind as Euler's, enunciate the theorem in 

 the convenient form 



a {n) 



- 2a (n - 1) - 2<r (n - 2) 

 + So- (n - 3) + Sa (n - 4) + 3<r (n - 5) 



-M?i-6) 



+ (-l) s - 1 sa(0) 



= 0, 



where o-(O) is defined to denote i(s 2 — 1); s being the coefficient 

 of a (0). 



The table of er (n) up to w = 3000, which was referred to at the 

 end of the last section, will be verified by this formula during its 

 passage through the press. 



§ 6. I have obtained also the following formula which is of 

 the same class. It does not however afford so complete a verifica- 

 tion of a table as that given in the preceding section, as certain 

 terms are omitted; and, further, as all the coefficients are +T 

 there is more chance of a compensation of errors. 



VOL. V. PT. 11. 8 



