j92 XV. Wahtno on fome Properties of the 
Cor. Hence S 7 (w) = S' ( m — /) 4- S / (w— 2/) — S ; (^ — 5/} — 
S' (m — 7/) 4 S 1 (in — 12/) 4 S'(m — <5/) — S 7 (/// — 2 2/) — 
S'(w - 26/) + &c. ; the law of the feries has been given in 
Cafe r. 
Cor. The fum of all the divifors of m not divifible by /== 
S (/») - S' (;»)=s S (m - 1) - S / (w — /) + (S (w — 2) — S' (02 — 2/)) 
— ,S (m - 5) — S 7 (w — 5 '^)) — (S (« - 7, - $ 7 (w — 7/ )) 4 &c. 
A fimilar rule may be predicated of the lum of the divifors 
not divifible by the numbers a, b, c , */, &c for the fum of 
the divifors of the number (m) divifible by a, b 9 c , ✓/, e, &c., 
where a , d, e , &c. are prime to each other = (S a (/«)4 
S* (02) 4 S' (/») 4 S J (m) 4 S < ( m ) 4 &c.)) - ((S aXb (jn) 4 S aXe (m) 4 
S* x<r (fl*)4 S aX< *(«J' 4 S &xd (m) +S cXrf {w) 4 S' x ' (m) 4 &c.)) 4 
(S' x * Xf (»«) S^ xixi (w) 4 (w) 4 S : XtX</ («) 4- S-+*+'( w ) + 
&c.)) — + S i+7 +‘+ # (02) 4 &c.)) 4 ((S*+* +f +^+'(tt) 
+ &c.j) - &c. = las the fum of all the divifors of m ... . divifi- 
ble by a, b , <*, d, e , &c. refpe&ivelv added together, - the fum 
of all the divifors of m divifible by the products {ab, ac, be , & c.) 
of any two of the quantities a , b , r, d, &c.-{-the fum of all 
the divifors of in divifible by the contents (abc, abd , acd , bcd 9 
&c.) of every three of the quantities a 9 b, c, d , &c. — the fum of 
all the divifors of m divifible by the contents of every four of 
the abovementioned quantities a 9 b , c , d \ &c. 4 and fo on, and 
confequently S (w) — C is the fum required. 
The principles given in the former parts may be applied to 
this, and extended to equations of which the factors have the 
formula — A; and from the fum of the inferior powers of 
each of the roots, and the co-efficients may be collected the 
fum of the fuperior; the fit me may be performed by the co- 
efficients only, &c. 
PART 
