Dr. waring on Interpolations. 63 
This problem may be demonstrated in the fame man- 
ner as the preceding theorems, by writing for a in the 
correcting quantity fucceffively its values ?r, <7, -t, &c. 
2. For the correcting quantity fought may be affumed 
the quantity 
x‘ -a‘ x X - e' X 
v — * * X * — ft X 
X S — y S XX s — y x &C. X X r X X s — §* XX s — cr S 
or' — yX or— b* X &C. X * X 7/ — / 
X Jtf* — r x &C. 
x T ' 4 + ^ 
X CL X x 
s s s s .s 0 r 
— @ X x —y X x — } X ccc . x x x 
X T \ &C. 
sss n isss*.sc> r s c 
? — a X? — £ X £ — 7 x ^ — 3 X OCC. X g x ? — n 
S S s S e 
# — ar X X — t x oCC. 
XT { + &C. 
Xj 
$ J O 
— T X ccc. 
3. In general, let z be any quantity which is = o,. 
when x becomes either a, ( 3 , y, J 1 , s, &c. : let z become 
fucceffively a, b, c, d, Sec. when x becomes ?r, g, a, r,. 
&c. refpeCtively. When x either = £>, c, r, See. let n = 0 ; 
but if a:=7r, let Yl-p: in the fame manner when x either 
= 7 r, cr, t, 8 cc. let P =0; but when x=g let P=r : and limi- 
larly, let 2 = 0 when x is either 7 r, £, r, &c. ; but when 
x = <r let 2 = s: and like wife, when x is either 7r, £>, cr,, Sec. 
let T =0; but when x=t let T —t: Sec. then for the cor- 
*Zj n 
reeling quantity fought may be affiimed — x - x T*+ 
Z p Z 2 Z T „ _ 
— x — x T ? + — X — x T + — X —x T + See, 
B T G S D > 
T H Er - 
