110 Mr Glaisher, On the sum of the divisors of a number. [Feb. 25, 



where a (0), when it arises, is defined to denote \n. For ex- 

 ample, 



a (10) - 3(7 (9) + 5a (7) - 7 a (4) + 9 x * ¥ ° = 0. 



§ 3. The following formula, which is of the same kind as 

 Euler's but in which n is restricted to be of a particular form, 

 was obtained by means of Elliptic Functions*. 



If n be of the form 8m + 7, then 



o- (n) - 2o- (ft - 4) + 2o- (ft - 16) - 2(7 (ft - 36) + ... = 0. 



For example, let ft = 55, the formula gives 



o- (55) - 2(7 (51) + 2o- (39) - 2<r (19) = 0, 

 that is, 72 - 2 x 72 + 2 x 56 - 2 x 20 = 0. 



In this formula a (0) cannot occur. 



§ 4. The formulae given in the preceding sections are in- 

 teresting, if only for the reason mentioned by Euler ; but for the 

 actual calculation of a table it is preferable to employ the 

 equation 



o-(ft) = o-(ft 1 )o-(ft 2 )o-(ft 3 ) .,., 



where n v ft 2 , ft 3 ..are prime to one another and n = n 1 n 2 n 3 .... 

 They would be useful in verifying a table of o- (ft), but, as the 

 intervals between the terms are unequal, the verification afforded 

 is not systematic, and it is not obvious how to apply them in order 

 to verify by their means all the numbers in a table. 



In seeking for formulas which would afford a more complete 

 verification of a table of a (n) I obtained also the following four 

 formulae : 



(i) 

 If ft be even, then 



C7 (1) (7(ft - 1) + (7(3) (7 (ft - 3) + (7 (5) (7 (ft - 5) . . . + (7 (ft - 1) (7(1) 



= |x sum of the cubes of those divisors of n which have uneven 

 conjugates. 



For example, putting n = 6, 



(7 (1) (7 (5) + a (3) (7 (3) + a (5) a (1) = 1(6 3 + 2 3 ), 

 that is, 6 + 16 + 6 = \ x 224. 



* Quarterly Journal of Mathematics, Vol. xx. p. 121. 



