solving the Problem of Interpolation. 461 
tions of a which I shall represent by ¢ (a) =u x (x) = v, 
v(x) = w; so that we have 
(1.) y=au+bv+cw++...... where a,b,c... are 
constant coefficients. 
Now, the question is, 1st, how many terms of the second 
member of the equation (1.) are to be employed, in order to 
obtain a value of y so approximate that the difference be- 
tween it and the exact value may be very small, and capable 
of being compared with the errors to which the observations 
are liable; 2ndly, to determine in numbers the coefficients of 
the terms retained, or, in other words, to find the approxi- 
mate value just mentioned. The data of the problem consist 
of a sufficient number of values of y represented by 
Yio Yoo veveeeees Yn3 
and corresponding with an equal number (7) of values of x 
Fepresented ‘by 7,,2,,...... Z,, and, consequently, with an 
equal number of values of each of the functions uw, v, w, ...... 
These several values of the functions I shall represent by 
Uy 9 Ug, seeeee U, for the function u; 
V,9 Uz 5. eeeeee, UV, for the function v; 
W 15 Woy seeeee Wy for the function w; &c. 
Thus we shall have for the solution of the problem, the num- 
ber (n) of equations of the first degree among the unknown 
coefficients a, b,c ...... 
(y= au.+ bu 4+ew + w.... 
(2.) Z Yo = BUly + DDy+ CWy+ cevece 
I e 
L Yn = AUgt Dg + C Wat cooces 
and if we put z to represent generally any one of the whole . 
numbers |, 2, ...... 2, these equations will all be comprised 
in the general formula 
(3.) Ye = AU; + DY; + CW; A cover. 
The first approximation will be made by neglecting the 
coefficients b,c, &c., or, what amounts to the same thing, by 
reducing the series, which the equation contains, to its first 
term. ‘Then the general approximate value of y will be 
(4.) Y= aw; 
and to determine the coefficient (a) we shall have the system 
of the equations 
(5.) Yy = Ay, Yo = Ay sevens Yn = Aty. 
The different values of a that can be deduced from the 
