552 
two smith’s prize dissertations. 
[551 
for clearly this is of the form in question A + Bx + Gx- + Da?, and y becomes = y 1 for 
x — a: = y 2 for oc=b, &c. And the like for any value of n. This is known as Lagrange’s 
interpolation formula. 
The given values of x may be equidistant, say they are 0, 1, 2,..., n— 1, and the 
corresponding values of y are y 0 , y 1} ..., y n _ x ; then writing down the expression 
where, as usual, 
&y 0 = yi-yo, & 2 y 0 = y. 2 -2y 1 + y 0 , &c. ; 
then for x = 0, 1, 2, &c. the values of y are 
yo, 
y 0 +&y 0 , =yi, 
y 0 + 2Ay 0 + A 2 y 0 , = y. 2 , 
&c., 
or the required conditions are satisfied. 
As regards the determination of the limits of error, taking a particular case n= 4, 
suppose that we have the values y 0 , y 1 , y 2 , y 3 of y corresponding to the values 
0, 1, 2, 3 of x, and that the true value of y is known to be 
= A + Bx + Gx 2 4- Dx 3 -f Kx A , 
where A is a function of x, which for any value of x within the given values (i.e. 
from x = 0 to x = 3) is known to be at least = P and at most = Q, i.e., K > P < Q, 
where to fix the ideas P and Q are each positive, Q being the greater. Here 
calculating the interpolation value of y — Kx i (the last term Kc? by Lagrange’s formula), 
we have 
x. x—1x — 2 
A 3 2/o 
1.2.3 
+ Ka? 
— % K 1 x{x — 2)(x— 3) 
+ 8 K<ix{x — l)(a' — 3) 
— %£-K 3 x(x — l)(x — 2), 
viz. this is the true value of y. Hence using the approximate formula as given by 
the first line, the last four lines give the error, viz. this is 
= Ko? + K 3 ^x 2 + if 2 (8^ 3 + 24x) + K x f ^ - K 3 {^a? +2ïx)+ K 3 (32x 2 ) - K x (±x 3 + Sx). 
But K x , K 2 , K 3 being each >P and < Q, this is 
> P (o?+ 8x? + 43îc 2 + 24&') 
— Q ( 14a? + S2x 2 + 30,r).