VOL. LXXVIII.] PHILOSOPHICAL TRANSACTIONS. 499 



ar*'-**, &c = a' = ; the sum of any power (m) of each of the roots in the 

 equation a' = O will be s' (m), where s' (m) denotes the sum of all the prime 

 divisors of the number m, and m is not greater than n. 



Cor. Hence, by the rule before-mentioned s'(ot) = s' (»i — 1) + s' (m — 2) 



— s' (w — 4) — s' (m — 8) + s' (m — 10) -|- s' (m — 1 1) — s' (w — 12) — 

 s' (m — l6) + s' (w — 17) + s' (w — 19) — s {m — 20) + s' (m — 23) — 

 2s' (m — 24) + s' (?» — 26) + s' (m — 27) — s' (w — 28) + s' (w — 29), 

 &c. 



If in this, or the preceding, or subsequent analogous cases s (m — r), or 

 s' (m — r), or s' (m — r), becomes s (O), or s' (o), or s^ (o) ; for s (O), or s' (O), 

 or s' (O), always substitute r. 



Cor. Let l be the co-efficient of the term x^'-'"; then, by the above-men- 

 tioned series contained in the Meditationes Algebraicae, will 1 .2.3.4.... 



wi X L = 1 — m . — - — s' (2) 



X s' (3) — m . w — 1 . m — 2 . "-^-^ X s' (4) 



+ m.m- 1 .m— 2. ^-=-^ x s' ((2))' 



— &c. be an equation, which expresses a relation between the prime divisors of 

 the numbers 1, 2, 3, 4 ... w — 1, »»> and their powers. 



Cor. The co-efficient l = the difference between the two respective numbers 

 of different ways that m can be formed by adding the prime numbers 1, 2, 3, 5, 

 7, ,11, 13, 19, &c. the one with, and the other without, 2. 



Part 3. — 1. Let an equation ar^* — 1 . -r^ — I . x^ — 1 . x^ ^ 1 X &c. = 

 x^ — px''-^ -f- qx^-'^ — rx^-^ -f &c. = O; then will the sum of the (m) 

 powers of each of its roots be the sum of all the divisors of m, that can be found 

 among the numbers «, p, y, <?, &c. 



2. The co-efficient of the term x^-'" will be the difference between the two re- 

 spective numbers of different ways, that the number (m) can be formed from the 

 addition of the numbers a, j3, y, J^, &c. ; the one containing in it an odd number 

 of the even numbers contained in a, (3, y. <?, &c. ; the other not. 



Part 4.— -L Let a?' — 1 . ar^' — 1 . x^' -^ I . x^ — 1 a?«^ — 1 . &c. = 



aJ> — px^-' -f qx^-^— ra^-3' + &c. = a^ — a?^-^ — a?*-*'-|- a^-s-'-f- oc^-ii 

 __ a:^-"^ — a7*-»5^ -|- &c. = B = O, of which equation all the co-efficients are 

 the same as in case the first, and consequently + 1 or to the term {x^-"'). \ 



2. The sum of any power I X m of each of the roots of the equation b = 

 will be s' (m) ; where s' (m) denotes the sum of the divisors of m, which are di- 

 visible by /. 



Cor. Hence s^ (m) = s' (w -- /) + s' (m — 2/) — s' (m — 5/) — s'(m — 7 1) 



3 s 2 



