56 Dr. Glaisher on Arithmetical Recurring Formula, 



all of which cancel one another ; i. e. there are five l's and five 

 (-l)'s, four 2's and four (-2)% four 3's and four (-3)'s, 

 three 4's and three (— 4)'s, two 5's and two (— 5)'s, one 6 

 and one — 6, one 7 and one — 7, one 8 and one — 8, one 9 

 and one —9. 



§ 6. As a second example let n=10, so that g is 4. We 

 form, as before, the groups 



4, 5, 7 



3, 9 3, 4, 6 



2, 4, 10 2, 8 2, 3, 5 



1,2,5,10; 1,3, 9; 1,7; 1, 2, 4, 



0, 2, 8 0, 6 0, 1, 3 



-1,5 -1, 0, 2 



-2, -1, 1 



and, changing the signs in the second and fourth groups, we 

 obtain the numbers 



-4, -5, -7 



3, 9, -3, -4, -6 



-2, -4, -10, 2, 8, -2, -3, -5 



1,2,5,10,-1,-3,- 9,1,7, -1, -2,-4, 



0, -2, - 8, 0, 6, 0, -1, -3 



-1,5, 1, 0,-2 



2, 1, -1 



all of which cancel one another except one —1, two (— 2)*s, 

 three ( — 3)'s, and four ( — 4)'s. 



§ 7. In the general theorem the actual divisors, and num- 

 bers formed from them, cancel one another (with certain 

 exceptions when n is a triangular number). We may there- 

 fore replace these divisors and numbers by any functions of 

 themselves, the functions being the same for all and changing 

 sign with the arguments : and we may further combine these 

 functions by addition, thus obtaining numerical theorems 

 which connect together functions of the divisors of the 

 numbers n, n — 1, n — 3, &c. 



§ 8. The simplest method of deducing a numerical formula 

 from the general theorem is merely to add together all the 

 numbers in each of the groups. It is evident that the sum of 

 the numbers in the first group is the sum of the divisors of n. 

 In the second group the sum of the numbers is equal to three 

 times the sum of the divisors of n — 1 ; for, d being any 

 divisor of n — 1, the numbers in the upper line are of the 



