Dr. Waring 01 1, &c, 389 
/S (m - 1) - 9S ( m - 2) + rS (jn ~ 3) - rS (w - 4) -f / S (w - 5) - 
&c = S (w - 1) 4- S {in - 2) - S (/« - 5) - S (/?/ - 7) 4- S (w - 1 2) 
+ S — 1 5) - S (Vw - 22) - S (m — 26) 4- &c. which is the pro- 
perty of tiie fum of divifors invented by the late M. Eulrr. 
Cor. By fubftituting for S (;w — 1), S(w — 2), &c. their 
values S (jn — 2)4-8 {m — 3) — S (/// — 6) — S(/;/ — 8) 4- &c., 
S (/« -3)4-8 (;;; — 4) — S (w - 7) - 3 {m - 9) 4- &c. &c. in the 
given equation S («) = S (ot — 1 ) 4- S (/// - 2) - S (/// — 5) 
— S (w— 7) + &c. may be acquired an ex predion for the fum 
S (77;) in terms of the fums of the divifors of numbers lei's 
than in — 1, in — 2, &c. — the fame method may be uied for 
a limiiar purpofe in fume of the following proportions. 
Cor. By tiie rule for finding the fum of the contents of 
every (>//) roots from the fums of the powers of each of the 
roots may be deduced the equation — 1 .2.3.4...;;/, 
or 0 — 1 - m . 0^2) 4- m . rn— 1 . — — 0 (3) 
2 / 3 
- tn . in — \ . m - 2 . — — S (4) 4- 6cc. 
4 
4-;/; .in — 1 . . f lZ -2 S ((2)) 1 - &c. 
in which the fum of the divifors of any number m is expreffed 
by the fums of the divifors of the inferior numbers in— 1, 
m — 2, See. and their powers. If v be an even number, then 
db 1 . 2 . 3 . . m will have the fame fign as the co- efficient; 
if uneven, the contrary ; but if the co-efficient = 0, then will 
the content 1 .2.3..;;/ vanifh. The law of this feries is 
given in the Meditationes Algebraical 
3. Let FI be the number of different ways by which the fum of 
any two numbers i, 2, 3, 4, . . . m - 2, ;// - 1, can become 
rr m ; FT the number of ways by which the fum of any three 
of the above-mentioned numbers can make m ; H ", H //y , IT" 7 , 
Sc c. the number of ways by which the fum of any four, five, fix, 
F f f 2 See. 
