Dr. waring on Interpolations. 
THEOREM, 
Aflame '(«) quantities a, (3, y, 3, t, Sec. then will the 
fum of all the (n) quantities of the following kind 
if 
+ 
u — (3 Xcc — yXa — — tX &C. |3 — X /3 — yX|3 — ^X/3 — «X &C. 
•v 0 
y— aXy — (^Xy — o'Xy — t X ScC. o' — a, X <2 — @ X d — y X o' — eX &C* 
+ Sec. = o, if m be any whole 
t — flftXs — ]3xt — yXt—8x &c. 
number lefs than n — i; but if m — n- i, then will the 
above mentioned fum = i . In general, the fum of the 
n terms a &c. _|_y^ £ &c. -f~<k c *) ^ 
a — i^Xa — y X a — ci'Xc*— £X &-C. 
/ 3 W (ay^&c. + ay £ &c.-l-a^&c.-!-ykVc.+&:c ) _|_ 
/3— a X /3 — y X 0— $ X /3— -e X &C. 
y (cc /3 $ &C. -f- a $ £ &C« occli &C. -{- &C. &C.) 
y — a X y — 0 X y — ^ X y — £ &C. ; 
^ ( a f3y. &C. -j- a.y e &c # -{- &c») ^ 
^ — a X £ — /3 X ^ — y X ^ — £ X &C. 
£ (a|3y &C. -j- a/3^&C. -{- «y^&C. -j- $y^& C. -j- &C.) f 
- ~ F —— ■ ■ 1 . - 7 i" 5 £C. — 
£— aXe— /3 Xe— yXe—^X &C. 7 
if w be lefs than ?z, and »z+r not equal to n- 1 , where r 
is equal to the number of letters contained in each of the 
contents above mentioned jSyJ', 8cc. (Sys, See. (3fo, Sec. 
y$t, Sec. Sec. See. refpe&ively : but if m+r-n- 1 , then 
will the above mentioned fum = ± i ; it will be + i if r 
be an even number, otherwife - i . 
4 
DEMON- 
