Dr. Glaisher on Arithmetical Recurring Formula?. 55 



where 1, d 1? d 2 , . . . , d, are all the divisors of r, the suffix 

 of G ; then the numbers given by the formula 



&n{d)-Gtn-i(d } d±l) + G n - 3 (d, d±l, d±2) 



-G n - 6 (d, d±l, d±2, d±3) + &c. 



all cancel each other if n is not a triangular number ; but 

 when n is a triangular number, i#(# + l), there remain 

 uncancelled 



one 1, two 2\s ; three 3's, . . . , g g% 



these numbers having the positive or negative sign according 

 as g is uneven or even. 



§ 5. As an example of this theorem, let n = 9. We write 

 down in a central line the divisors of 9, 8, 6, and 3. In the 

 line above, beginning with the divisors of 8, we write the 

 numbers obtained by adding unity to the divisors, and in the 

 line below the numbers obtained by subtracting unity from 

 the divisors : in the second line above, beginning with the 

 divisors of the next number, 6, we write the numbers obtained 

 by adding 2 to the divisors, and, in the second line below, the 

 numbers obtained by subtracting 2, and so on. The scheme 

 of numbers thus formed is 



1, 3, 9 ; 



We then change the signs of all the numbers in the second 

 and fourth groups, thus finally obtaining the system of 

 numbers 









4,6 







3, 4, 5, 8 



3, 5 



2, 



3,5,9 



2, 3, 4, 7 



2,4 



1, 



2, 4, 8; 



1, 2, 3, 6 ; 



1,3. 



o, 



1,3,7 



0, 1, 2, 5 



0,2 







-1, 0, 1, 4 



-1,1 



-2,0 



1, 3, 9, 









-4, 



-6 







3, 4, 5, 8, 



-3, 



-5 



2,-3, 



-5, 



-9, 2, 3, 4, 7, 



-2, 



-4 



•1, "2, 



-4, 



-8, 1, 2, 3, 6, 



-1, 



-3 : 



0,-1, 



-3, 



-7, 0, 1, 2, 5, 



o, 



-2 







-1, 0, 1, 4, 



1, 



2, 



-1 







