Dr. Glaisher on Arithmetical Recurring Formula. 57 



form rf+1, and those in the lower line of the form d — 1, so 

 that the sum of the three lines is 3 J£<d. In the third group 

 the sum of the numbers is five times the sum of the divisors 

 of n — 3; and so on. 



We thus obtain as result that 



ff(n)-3o-(w-l) + 5 <r(w -3)-7<r(n-6) + 9 or(n- 10) -&c. 



is equal to zero if n is not a triangular number, and is equal to 



(_l)^-i(12 + 2 2 + 3 2 + ...+/) ? 



when n is the triangular number ^g(g + l). 

 We know that 



l» + # + a»+,..+p»:=tofo + l)(Sfr+l) J 



and since the coefficient of er(0), when it occurs, is 

 ( — 1)^(2^ + 1), we may evidently equate the <x-expression 

 to zero, for all values of n, if we put o-(O) equal to ig(g+l), 

 that is, equal to Jw. 



§ 9. Since the general theorem is merely concerned with 

 the mutual cancellation of a system of numbers, i. e. since it 

 merely asserts that a number which occurs any number of 

 times with the positive sign will occur exactly the same num- 

 ber of times with the negative sign (except in the case of 

 certain numbers, when n is a triangular number), it is evident 

 that we are at liberty to change the signs of all the even 

 numbers throughout (or, indeed, the signs of all the numbers 

 of any particular form). 



Changing the signs of the even numbers, and adding 

 together the numbers in each group, we evidently obtain 

 f (n) from the first group. The sum of the numbers in the 

 middle line of the second group is f(n — 1), and the sum of 

 the numbers in the upper and lower lines is — 2£(n — 1). 

 Similarly in the third group the sum of the numbers in the 

 middle line is £(n — 3), the sum of the numbers in the next 

 upper and next lower lines is — 2J(n — 3), and in the highest 

 and low r est lines is 2f(w — 3) ; and so on. 



Finally, changing the signs of the alternate groups, we 

 obtain the expression 



f(n) + S(ii-l) + C(n-3) + £(n-6) + Kn-10) + &o. 



When the signs of the even numbers are changed, the 

 numbers which remain uncancelled in the general theorem 

 are 



(-l)*- ! {one 1, two (-2)'s, three 3 V . . , 9(±9Y*h 

 Adding these numbers we obtain the series 



, (-l)?-i{l2-2 2 + 3 2 -4 a + ...+ (-l)*-y} J 



