Dr. waring on Interpolations. 61 
fpe&iveiy s“, s s , s y , s J , s e , &c.; then will y = 
x r X X s — @ S X X s — y X X S — $ S X X* — s* X &C. 
— — — . ^ g-* 
«' x &c. 
x r x X s -a 5 x.v J -y'xr- of 1 — a" X &C. a 
4“ ■■ — — . - — ^ gp 
@ r — a S Xi 2 s — y s X @ s — X — s s x &c. 
x r X X s — a. S X X s — @ S X X 5 — 0 S x X s — e J X &C. v 0 
y r x y s — * 1 x y -1 — @ 5 x ■/— Xy J - £* x &c. 
This may be demonftrated in the fame manner as 
the preceding theorem, by writing a, (3, y, o, e, &x. fuc« 
ceffively for x. 
PROBLEM. 
Let there be n values x, j0, y, 8ec. of the quantity x r 
to w r hich the n values s®, s B , s’q s*, s E , 8ec. of the quan- 
tity y correfpond; fuppofe thefe quantities to be found 
by any fun<5tion X of the quantity x; let 7 iy £>,. a, t, &c. 
be values of the quantities x, to which s’ 1 , sq sr, s T , &c. 
values of the quantity y correfpond : for x fubftitute its 
abovementioned values tt, v, r, &c. in the function X, 
and let the quantities refulting be s v , s?, s’ 7 , s T , Sec. not 
equal to the preceding s T , sq s', s% 8cc. refpectivel y ; to 
find a quantity which added to the function X fhall not 
only give the true values of the quantity jy correfponding 
to the values oc } (3, y, §, s, Sec. of the quantity x, but alfo 
cor- 
