551] 
551 
551. 
TWO SMITH’S PRIZE DISSERTATIONS. 
[From the Messenger of Mathematics, vol. n. (1873), pp. 145—149.] 
Write dissertations on the following subjects : 
1. The theory of interpolation, with a determination of the limits of error in the 
value of a function obtained by interpolation. 
2. Determinants. 
1. The general problem is to find y a function of x having given values for 
given values of x. The problem thus stated is of course indeterminate; in practice, we 
assume a certain form for the function y, the coefficients of which form are determined 
by the given conditions, viz. either y is known to be of the form in question, the 
actual value being then determined as above, or it is assumed that y is approximately 
equal to a function of the form in question, and the value is then approximately 
determined in such wise that, for the given values of x, the function y shall have its 
given values. 
The ordinary case is when we have the values of y corresponding to n given 
values of x, and y is taken to be a function of the form A + Bx + ... + Kx n_1 . 
Suppose to fix the ideas n= 4, and that y lt y 2 , y s , y i are the values of y corre 
sponding to the values a, h, c, d of x. We may at once write down the expression 
(x — h) (x — c) (x — d) 
y * (a — b) (a — c) (a — d) ^ 
+ 
(x — c) (x — d) (x — a) 
0b-c) (b-d) (b -a) 2/2 
(x — d) (x — a) (x — b) 
+ (c — d) (c — a) (c — b) 
V3 
(x — a) (x — b) (x — c) 
+ (d-a) (d-b) (d-c) 
2/4,