466 M. Cauchy on a New Formula for 
Ay, can be compared with the errors of observation, we may 
disregard A y and reduce the approximate value of y to 
aSy;. 
In the contrary case we shall determine 6 by means of the 
formule 
(IV.) v=aSv,+ Av Av=BS'Azy,, 
(S'A v; being the sum of the numerical values of A v;,) and the 
difference of the second order (A* y) by means of the formula 
(V.) Ay=BS'Ay + A*y. 
If the particular values of A?y represented by A*®y,, A? Yo» 
. A*y, may be set off against the errors of observation, we 
shall be able to neglect A*y, and therefore to reduce the ap- 
proximate value of y toa Sy; + CS’ A y;; but if they can- 
not, we shall determine y by means of the formule 
(VI.) w=aSv;,+ Aw, Aw= BS Aw, + A®w, A?w 
= y S! A’ w;. 
(S" A?w,; being the sum of the numerical values of A? w;,) 
and the difference of the third order (A® y) by means of the 
formula 
(VII.) A?y = yS" A®y; + A®y, &c. 
Thus, in short, by supposing the coefficients «, 6, y ... de- 
termined by the system of the equations (II.), (IV.), (VI. \ &e. 
we shall calculate the several orders of differences represented 
by Ay, A? y, A°y...... or, rather, their particular values cor- 
responding with the values (7, 7, 73...) of the variable z, 
until we arrive at a difference the particular values of which 
may be set off against the unavoidable errors of observation. 
Then it will be sufficient to represent as zero the value of this 
difference deduced from the system of the equations (III.), 
(V.), (VII.)... in order to obtain a sufficient approximation to 
the general value of y. 
This general value will be then 
y=aS8y;,, or y= aSy,+ BS Ay;... &e. 
according as we shall be able, without a sensible error, to re- 
duce the series (I.) to its first term or its first two terms, &c. 
Now, if we call the number of terms retained m, the problem 
of interpolation will be resolved by the formula 
(VIIL) y= aSy,+ BS’ Ay; + yS" A’y; + &e., 
the second member being continued to the term which con- 
tains A"—1y;. 
It is necessary to observe, that from the formule (II.), (III.), 
(1V.), (V.), (VI.) (VIL.) ... we derive not only 
