solving the Problem of Interpolation. 465 
(18.) NY, == BiG. 
Let A®y be the remainder of the second order which is re- 
quired to complete this approximate value, and let us there~ 
fore put 
(19.) Ay = BS'Ay, + Ary. 
Let us in like manner put 
(20.) Aw = BS' Aw; + A’w, &e. ... 
We shall derive successively from the formula (16.) 
(21.) Ay; = bAv,+ chw; + &e. ... 
(22.) SAy; = bS9’Av,;+ cS Aw,, &e. ». 
and from this last, multiplied by 6 and deducted from the 
equation (19.), we obtain 
(23.) Aty =cA?w +, &C. oe 
Let 6;, A®y;, A°w;...... be what the values of 6, A?y, A? w. 
derived from the equations (17.), (19.), and (20.), become when 
for x we substitute z;, i being one of the integers 1.2... 2. 
If the values of A? y,, A? y, ... A?y, be very small, and capa 
ble of being compared with the errors incident to the obser- 
vations, it will be useless to proceed to a new approximation, 
and we may be contented with the approximate value of Ay 
furnished by the equation (18.). If it happen otherwise we 
shall obtain a third approximation by operating upon the 
formula (23.) as we have done in the first approximation on 
the formula (1.). By continuing this process we shall obtain 
the following rule. 
The unknown quantity y, a function of the variable quan- 
tity x, being supposed capable of being developed in a con- 
verging series 
aw+buv+cewt. 
in which w, v, w represent given Fanelians ea the same variable, 
if we know x particular values of y corresponding with  par- 
ticular values (7,, v5, v3,.-. ,) of x: if moreover we represent 
by 7 any one of the whole numbers 1,2, 3.00,” and by ¥;; 2%, 
V;-+» what y, u, v, ... become when for z we substitute x;; then, 
in order to obesin a sufficient approximation to the general 
value of y, we shall first determine the coefficient « by. means 
of the formula 
(II.) u=aSu;, 
(in which S uw, represents the sum of the numerical values of 
u,;,) and the difference of the first order Ay by means of the 
formula 
(IIT.) y=aSyt+ Ay. 
If the particular values of A y represented by 4 my » AYy oe 
Third Series. Vol. 8. No. 49. June 1836. 
