Sum of the Divifors of Numbers. 391 
— &c. be an equation, which exprefl'es a relation between the 
prime divifors of the numbers I, 2, 3, 4 . . . m - 1, m , and 
their powers. 
Cor. The co-efficient L = the difference between the two 
reipedfive numbers of different ways that m can be formed 
adding the prime numbers 1, 2, 3, 5, 7, 11, 13, 19, 6cc. 
the one with, and the other without, 2. 
PART III. 
1. Let an equation x n - 1 . x & - 1 . - 1 . x* - 1 x &cc. 
x h — px b - 1 -\-qx b ~ z - rx b ~ 3 + &c. = o ; then will the fum of the 
(m) powers of each of its roots be the fum of all the divifors 
of m, that can be found amongff the numbers a, f 3 , y, See. 
2. The co-efficient of the term x h ~~ m will be the difference 
between the two reipedtive numbers of different ways, that 
the number ( m ) can be formed from the addition of the num- 
bers oc, ( 3 , y, 3 , &c. ; the one containing in it an odd number of 
the even numbers contained in a, (3 , y , &c. ; the other not. 
PART IV. 
1. Let x l — 1 . .v 1 — 1 . x * 1 — 1 . x * 1 — 1 x nl — 1 . &c. = 
x b — px b ~ ! 4- qx b ~ z] — rx b ~^ + &c. =* x b — x'"~ l — x b ~ l! + x h ~ s l + x h ~' 71 
— x h ~ lzl — x b ~ r5/ + &c. = B = o, of which equation all the co- 
efficients are the fame as in cafe the firff, and confequently 
±1 or o to the term 
2. The fum of any power / x m of each of the roots of the 
equation B = o will be S l (m), where S 7 ( m ) denotes the fum of 
the divifors of m, which are divifible by l. 
Cor 0 . 
