498 PHILOSOPHICAL TRANSACTIONS. [aNNO 1788. 



be s (m)f where s (m) denotes the sum of all the divisors of the number m, if m 

 be not greater than n. 



Cor. Hence (by the rule for finding the sum of (m) powers of each of the roots 

 from the sum of the inferior powers and co-efficients of the given equation) may 

 be deduced s (w) = jbs (m — l) — ^s (m — 2) + rs (w — 3) — 5S (m — 4) -f 

 is (m — 5) — &c. = s (w — 1) -f s (m — 2) — s (m — 5) — s (m — 7) + 

 s (m — 12) + s (w — 15) — s (»J — 22) — (m — 26) -f &c. which is the 

 property of the sum of divisors invented by the late M. Euler. 



Cor. By substituting for s (w — l), s (m — 2), &c. their values s (m — 2) -f- 

 s (tw — 3) — s (m — 6) — s (m — 8) + 8fC., s (m — 3) + s (m — 4) — 

 s (w — 7) — s (m — 9) + &c. &c. in the given equation s (m) = s (m -— l) 

 4- s (m — 2) — s (m — 5) — s (m — 7) + &c. may be acquired an expression 

 for the sum s (m) in terms of the sums of the divisors of numbers less than m—l, 

 m — 2, &c. : the same method may be used for a similar purpose in some of the 

 following propositions. 



Cor. By the rule for finding the sum of the contents of every (m) roots from 

 the sums of the powers of each of the roots, may be deduced the equation + 



1 . 2 . 3 . 4 . . . »w, or = 1 — m . — .— s (2) + w . m — 1 . — —: s (3) 

 — ffi.wi — l.m — 2. -~~ s (4) + &c. 



+ ,n.m-l.^-^ .^^ s ((2))=- &c. 



' in which the sum of the divisors of any number m is expressed by the sums of the 

 divisors of the inferior numbers m — 1, m — 2, &c. and their powers. If i* be 

 an even number, then + 1 . 2 . 3 . . m will have the same sign as the co-effici- 

 ent ; if uneven, the contrary ; but if the co-efficient := O, then will the content 

 1 . 1 . 3 . . m vanish. The law of this series is given in the Meditationes Alge- 

 braicsB. 



3. Let H be the number of different ways by which the sum of any two num- 

 bers 1, 2, 3, 4, ... m — 2, m ■— If can become = w ; h' the number of ways 

 by which the sum of any 3 of the above-mentioned numbers can make m ; h", h'", 

 h"", &c. the number of ways by which the sum of any 4, 5, 6, &c. of the above- 

 mentioned numbers is = m respectively ; then will 1 — h + h' —• h'" -}- h''" — 



&c. = + 1 or O. Let m = — — , and it will be -f 1 or — 1, according as 



z is an odd or even number, in all other cases it will be = O, 



Part 2. — 1. Let the equation be a? — I , x^ •— I . a^ — 1 . a?* — 1 . a?' — 1. 

 flp" — 1 . a?'^ — 1 . a?''— 1. . .a?"— 1 . &c. = x^' — px^'-^ + gx^'-'^—rx^'-i-^sx'''-* 

 — &c. = a?*' — x^'-^ — a,'*"'-* 4- x^-^ -\- x^'-^ — a?*'-*° — a;^'-" -f a^'-" -f- 



