500 P.HILOSOPHICAL TRANSACTIONS. [ANNO IJSS. 



-I- s^ {m — 12/) -\- s' {m — 15/) — s' {m — 111) — si (m ^ 20/) -f &c. ; the 

 law of the series has been given in case 1 . 



Cor. The sum of all the divisors of m not divisible by / = s (m) — s^ (m) = 

 s (^m — 1) — s^ (m — /) + (s {m — 2) ~ s' (m — 2/)) — s (m — 5) — s^ (m— 5/)) 



— (s (wi — 7) — s' (m — 7/)) + &c. 



A similar rule may be predicated of the sum of the divisors not divisible by the 

 numbers a, b, c, d, &c. : for the sum of the divisors of the number {m) divisible 

 by a, b, c, d, e, &c., where a, b, c, d, e, &c. are prime to each other = (s"(m) + 

 s* (m) + s' (m) 4- s'^ (m) + s^ (m) + &c.)) — ((s^x* {m) + s^x^ {m) + s*x^ 

 (w) + s^x^ (^m) + s^x^ (jn) _{_ g^x^ [m) + s^x« (m) + &c.)) + (s-^x^xc (^j^) — 

 S'X^X^ (w) + S-'XAX-/ (yre) _|- s^XfX^ (;^) + 8^^+*+^ (7/z) + &c.)) — ((s^+^+^+^ {m) 



+ s^+*+^+' (w) + &c.)) + (s^+*+^+'^+^ {m) + &c.)) — &c. = / = the sum of 

 all the divisors of 7?z . . . . divisible by a, b, c, d, e, &c. respectively added together, 



— the sum of all the divisors of m divisible by the products (a6, ac, be, &c.) of 

 any 2 of the quantities a, b, c, d, &c. -j- the sum of all the divisors of w divisible 

 by the contents {abc, abd, acd, bed, &c.) of every 3 of the quantities a, b, c, d, &c. 



— the sum of all the diyisors of m divisible by the contents of every 4 of the 

 above-mentioned quantities a, b, c, d, &c. -\- and so on, and consequently s (m) 



— c is the sum required. 



The principles given in the former parts may be applied to this, and extended 

 to equations of which the factors have the formula ar« '^h ; and from the sum of 

 the inferior powers of each of the roots, and the co-efficients, may be collected 

 the sum of the superior ; the same may be performed by the co-efficients 

 only, &c. 



Part 5. — 1. S (a X P) = « X s ((3) + sum of all the divisors of j3 not di- 

 visible by a = (3 X s (a) -|- sum of all the divisors of u not divisible by (3. 



2. S^ (a X |3) = «. X s^ (|3) + sum of all the divisors of |3 divisible by / but 

 not by a = &c. 



3. S(xXPXyX<yX &c.) =«Xs((3XyX^X£, &c.) + sum of all 

 the divisors of |3 X y X <^ X f, &c. not divisible bya = aX|3X s(yX^X£, 

 &c.) -|- sum of all the divisors of (3 X y X ^ X £, &c. not divisible by a -|- a X 

 sum of all the divisors of y X ^ X s, &c. not divisible byp = aX(3XyXs 

 {$ X £, &c.) + sum of all the divisors of p x y X ^ X f, &c. not divisible by 

 a 4- a X sum of all the divisors of y X <? X £;, &c. not divisible by p -j- « 

 X P X sum of all the divisors of $ X h &c. not divisible by y = « X P 

 X y X ^ X s (f, &c.) + sum of all the divisors of p X y X cJ' X f, &c. not di- 

 visible by a -I- « X sum of all the divisors of y^i, &c. not divisible by p -|- a x P 

 X sum of all the divisors of Ss, &c. not divisible byy + aXPXyX sum of 

 all the divisors of £, &c. not divisible by ^ = &c. The law of the series is mani- 

 fest. The letters a, p, y, i, &c. which are not contained between the parentheses, 

 denote prime numbers. 



