MISCELLANEA. 
I. Some Notes on Interpolation in n-dimension Space. 
By AV. PALIN ELDEKTON. 
1. IiUroductorjj. The main idea of this papei' is summed up in the principle, or the extension 
of the principle, that in certain cases the use of linear second dififerences will give a better result 
than the use of two-variable first differences, e.g. three points in the table chosen on a straight 
line may give a more accurate interpolation than the four "nearest" points. The usual formulae 
of interpolation assume that we require to find from tabulated values of a value of the 
function corresponding to a certain value of x not among the tabulated cases. In some circum- 
stances we have to deal with a function of two variables, and it was recently shown by 
Mr John Spencer* that in certain circumstances the simi>le one-variable interpolation formulae 
could be used. This saves a great deal of work in practice, and in the examples dealt with it 
was found to give accurate results. The object of the present paper is to supply a general 
method and to show that it is not necessary to limit the number of variables to two. Before 
proceeding to the subject itself it will be well to outline the alternative methods available. 
The most usual practice in two-variable interpolation is to interpolate between four values ; 
thus if is required and ?<i,n, u^, Uio and Un are given, then 'U,.o is found from Uqq and Mio, 
and Vyi from and u^i, and then ((,.5 from tOyo and u,.i. Graphically these processes might 
be represented by the following diagram, in which the dots show the positions of the given 
values, the crosses the positions of the interijolated values, and the lines connecting the points 
indicate the values used in each interpolation. Of course to give the values in a figure we 
should have to erect i^erpendiculars to the surface of the paper. 
2/ = l • • 
y = 0 
x = Q 
Fig. 1. 
The general solution in terms of differences is obtained by expanding the right-hand side of 
!<,,, = (l-|-Aj'-(H-A„)-'tf,„„ 
* John Spencer, "Some practical bints on two-variable interpolation," t/oH/'«ai of the Institute of 
Actuaries, Vol. xl. pp. 293 et seq. 
