390 j D r. Waring on fame Properties of the 
&c. of the above-mentioned numbers is = m refpedtively ; then 
will i ~H+H' - H" + H'" - &c. - + i or o. Let m= i?— , 
2 
and it will be -f i or — i, according as z is. an odd or even 
number, in all other cafes it will be=o. 
PART II. 
i. Let the equation be x — x . at — i . x* ~ i . x* — i . x 1 - i * 
.v 11 — i . x 13 - i . a .' 17 — i ... x ’ 1 — i . &c. = x b ' — px h ~~ l 4 qx b ~ z 
— r,x b '~ 3 4 sx b ~ 4 — &c. x h - x h ‘~ l — x b ~~ 2 4 x °' _4 -{- A? y—S — a ,6 ’~ 10 
— 11 + x b ~ lz 4- a 4~ i6 - aA~ 1 7 — x b ~^ 4 x b 20 — a : 6 ’ - 2 3 4 2 a; 6 ' -24 
1 
— x b '~ 26 — v 4 ’ -27 + x b '~ z ^ 9 &cc. = A / — • o ; the fum of any power 
(iw) of each of the roots in the equation A 7 = 0 will be S 7 (t»), 
where S 7 (772) denotes the fum of all the prime divifors of the 
number m, and m is not greater than n. 
Cor. Hence, by the rule before-mentioned S 7 (m) = S ' (m — 1) 
4 S' (rn - 2) — S 7 (m - 4) — S 7 (rn - 8) 4 S 7 (» - ic) + S 7 (w- 1 1 ) 
— S 7 (772 — 12) — S 7 ( m - 16) 4 S 7 (m — 1 7) 4 S 7 (772 — 19) — 
S 7 (w — 20) 4 S 7 (w - 23) — 2S 7 (772 - 24) 4 S ' (m -26) 4 
5 7 ( m - 27) — S 7 (m - 28) 4 S 7 (772 - 29), &c. 
if in this, or the preceding, or fubfequent analogous cafes 
5 (m - r), or S 7 (jn - r), or S l (m - rf becomes S (of or S 7 (of 
or S^) ; for S {of or S 7 (of or S 1 (of always fubftitute r. 
Cor. Let L be the co-efficient of the term x b ~~ m ; then, by 
the above-mentioned feries contained in the Meditationes AL- 
gebraicae, will 1 . 2 . 3 . 4 .. . . m x L = 1 - in . T h— - S' (2) 
X S 7 ( 3 ) — 772 . 772—1 . 772 — 2 . 'C—C X S 7 ( 4 ) 
X S'((2))* 
- &c. 
4 772 . 772 — I . 
4 &C. 
rn — 1 
4 772 . 772 - I . m - 2 . 
4 
*»- 3 
S 
o 
J 
