60 Dr. waring on Interpolations. 
let x, (3, y, S, e, See. denote n values of the unknown 
quantity x, whofe correfpondent values of y let be re- 
prefented by s*, s' 5 , s v , s'\ s', See. Then will the equa- 
tion a + b x+c x z + d + e x 4 
x—@Xx — yXX—$XX — eX &c. 
ee — iSX 54 — 7X a ~ s X &C, 
x n ~ 1 -y 
xS* + 
&c. ^ c(3 
X &c. * 
# — a x # — — £ X &c. «*■- 
= = xs r +f 
& X * — & X x — y X x — e X &C. 
y— a xy — 0X y — £ X 7— £ X &C. 
« X ^ — @ X $ — yX ^ — e X &C, 
x S c 
X — ce X x — (3 X x — y X X — <S X be. 0 
+ =l— = — -= j— rr= X S' + &C. 
i-eXi-lSx t — yXe — Sx &C. 
DEMONSTRATION. 
Write a for in the equation _y = 
* — /3 X X — yXX — ^XX — eX be. 
a — @X<x,— yX&— ^Xa — fX &C. 
x S* + 
x — »x x — yX x 
x be. 
@ — oox(2—yx(3—bx@ — sX be. 
x S^-f 
\ 
Sec. ; and all the terms but the firfl: in the refulting equa- 
tion will vanifh, for each of them contains in its nume- 
rator a factor x—ol—x—ol—o\ and the equation will be- 
come _>’= 
«-(?X«-yXa-iXa--iX be. 
oc — fix Oi—y X cx. — $ x Oi — e X be. 
X s a =s*. 
In the fame 
manner, by writing /3, y, S, e, 8 ec. fucceffively for x in 
the given equation it may be proved, that when x is 
equal to /3, y, S, i. See. then will y become refpeciiively 
$0, s v , s*, s', which was to be demonftrated. 
2 . Affume y=ax r +&x r + s +cx r + 2s +dx r +3 s . . . x r +^~~ ls '. 
and when x becomes oc, (3, y, S, s, See. let jr become re- 
fpedtively 
