58 Dr. Glaisher on Arithmetical Recurring Formulae. 



the sum of which is ^g(ff + ^-). We may therefore equate 

 the above ^-expression to zero for all values of n, if we put 

 ?(0)=-«. 



§ 10. In thus deriving the two formulas from a common 

 origin we obtain, so to speak, an arithmetical reason for their 

 general similarity of form, and for the dissimilarity in their 

 coefficients : in fact, we see that the one result arises in the 

 form 



«r(n)-(l + 2)o(n-l) + (l+2 + 2)<j(»-.3) 



-(l + 2 + 2 + 2>r(n— 6) + &e. 



= 0or(-l>- 1 {l 2 + 2 2 + 3 2 + ...-!-/}, 

 and the other in the form 



?(/0-(l-2)?(^-l) + (l-2 + 2)r(n-3) 



-(l-2 + 2-2)?0-6)+&c. 

 = 0or (-l)^-Ml 2 -2 2 + 3 2 -... +(-l) g -y\. 

 § 11. Between the a- and ^-formulae which involve as 

 arguments all the numbers from unity to n, the resemblance 

 is even closer. 



The c-formula may be written 



o-(n)-2o-(w-l)-2o-(/i-2) + 3o-(w-3) + 3o-(w-4) 



+ 3cj(n~5)-4cr(?i-6)-4(7(^-7)-... + (-l)p- 1 po-(0) = 0, 



where cr(0) is to receive the conventional value ^(p 2 — 1), 

 p being defined as the coefficient of cr(0). 

 The corresponding f-formula is 



J(n) + 2-{Xn-l)+2{:(n-2)+3{:(n-3)+3C(n-4) 



+ 3?(n-5) + 4?(n-6) + 4f(w-7) + ...+pcr(0) = 0, 



where ?(0) has the value -|(> 2 -1) *. 



The two formulas have the same coefficients, i. e. the first 

 term has the coefficient unity, the next two have the coefficient 

 2, the next three 3, and so on ; and they differ only in the 

 signs of the groups with even coefficients, which are negative 

 in the <r-formula. The values assigned to a (0) and ?(0) are 

 the same in magnitude for the same value of n or p } and 

 differ only in sign. 



§ 12. As before, we find that both formulas maybe deduced 

 from a general theorem relating to the actual divisors of the 

 numbers n, n—l,n — 2, &c. by adding the numbers in the 

 groups, and by adding the numbers in the groups after chang- 

 ing the signs of the even numbers. 



* These formulae were given in Proc. Lcrad. Math. Soc. vol. xv. (1884) 

 pp. 118, 119; and Proc. Camb. Phil. Soc. (1884) p. 119. 



