60 Dr. Grlaisher on Arithmetical Recurring Formula?. 



§ 15. Adding the numbers in each group, we evidently 

 obtain the expression 



ff(7i)-2{ff(»-l)+o-(7i-2)}+3{<r(n-3) + o-(n-4) + o-(n-5)} 



-4[o-(n-6)+.... + <7(n-9)}+ &c; 



and the sum of the numbers which remain uncancelled is 



l 2 + 3 2 + 5 2 +... + (p-l) 2 

 or 



_{2 2 + 4 2 + 6 2 + ...+ (^-l) 2 } 



according as p is even or uneven. 

 Now, p being even, 



12 + 32 + 5 2 + _ # + ( p _ 1 )2 = l (i) 3_^ 



and, p being uneven, 



2 2 + -i 2 + 6 2 + ...+ ( j p-l) 2 = l(/- j p), 



so that, for all values of p, the cr-expression is equal to 

 ( — l)^(j9 3 — -p). It is easy to see that p, defined by the 

 fact that ^p(p + l) is the triangular number next superior to 

 w, is the same as the p of § 11, which was defined as the 

 coefficient of + o"(0) in the formula : we may therefore equate 

 the ©--expression to zero if we put c-(O) = ^(p 2 — 1). 



§ 16. Similarly by changing the signs of the even numbers 

 and then adding the numbers in each group we find that the 

 expression 



t(n)+2{?(n-l) + f(»-2)|+3{f(n-3)+?(n-4)+5(n-5)} 



+ 4{£(n-6) + ...+?(»-9)}+&q. 



is equal to 



l 2 + 3 2 + 5 2 + ... + 0-l) 2 or 2 2 + 4 2 + 6 2 + ... + (p-l) 2 



according as p is even or uneven. By replacing these series 

 by their sums, as in the preceding section, we obtain the 

 result in the form given in § 11. 



§ 17. The general theorems of §§ 4 and 13 were proved in 

 a paper which was communicated to the London Mathe- 

 matical Society last May.* They are there deduced from 

 analytical formulae connected with the Zeta Functions. A 

 purely arithmetical proof of them would be very interesting, and 

 it is probable that it would be easier to obtain such a proof for 

 the general theorems than for the recurring formulas which are 

 deducible from them by addition. 



There are various other general theorems relating to the 

 actual divisors which reduce to the recurring formulas of 

 §§ 1, 2, and 11 when the numbers in each group are added 



* u 



Relations between the Divisors of the first n natural numbers : 



read May 14, 1891. 



