66 
Dr. waring on Interpolations. 
X — 00 XX— y XX — $X X — EX &C. X — 0LXX — (3xx — %Xx — txk' 
B = , C — 
£ — » X (3 — y X @ $ X fi — £ X &c. 
— axy — 0Xy — $Xy — e X &C. * 
X— cc X X — 0 X X — ? X X — s X &C. X — ooXx — 0XX — yXX — 2x 6fC. 
^ J — C 6 x £ — 0 X < 3 '“ 7 X ^ — 5 X &C. * ^ H _a x e — /?X e — X =~ X & C. * 
Sec. : fubftitute thefe values for a, b, c, d, e, Sec. refpec- 
tively in the preceding equations (a + b + c + d + e + Sec. = i , 
Aa + B/3+Cy+D.(J + ES+ Sec. = Aaf-b b^S* h- cy 2 + 
E6 2 + Sec.^a; 2 , a& 3 + b/3 3 + cj/ 3 + d J V3 + Eg 3 4- Sec.^a: 3 , Sec.) 
and there refult the equations (i) 
X—@XX—yXX—$ 
£ X &C, 
— /3 X a— yX a. — u X cc — eX &C, 
^-aX,r- 7 Xx'-Jx^- £X &C. x—aXx— 0 Xx—i XX — t x 3 tc. 
— j , -_- ■■ — f- T- — r — — : — + XTC — 1* 
$ — ot, x (2 — y X0 — $ X 0— «X&c. y—xXy~0xy—$Xy—tX&ce. * 9 
x — 0X x— y XX — $Xx — bX See. 
OCX a __ a y X « — & X a — £ X &c. 
+/S 
X 
X — ooXx — yXx — ^XX — eX &C+ 
-<*X0~yX0-2x0—iX &c. 
X — GO XX ^X*'' £ X &C» r\ 
+ yx + 8ec. = at; 
/ y-aXy-^Xy-^Xy-fX&c. 
x-0Xx-yXx-#Xx-ex&c. 
{3)«- x a -(3x«-rX“-^'X“— ^&ci ‘ ^ x £-«xe-rxe'-*xe-txbc 
i ?j-*Xx-yXx-2Xx-‘Xkc. 
X 
+ y ! 
at — &X x 0 X x — &Xx — £ X &c. 
y~*Xy-0Xy 
=™=x^ = *’; and in g eneral > 
ec * x 
a—^Xx — yX oi — iX“ — £ X&c. 
P-*xe-yX0- *X p-tX&c. 
OT *-<sX*-ffX*-'?x*-iX&:c. + ^ , r-<*X*-ffX*-yX*-»X&c. 
y X y— »Xy — ®X7- &Xv— fX&c. S—kx 2 — @ xS— yX^~— ‘X&c. 
+ 8cc.=x ,n , whatever may be the values of the quantities 
at ; a, jS, y, s, &c. : reduce all thefe fractions into terms, 
proceeding according to the dimenfions of the quantity 
ivj and it is evident, that the fum of all the fractions mul- 
tiplied 
