Miscellanea 
95 
but it is, I think, better to use coefficients* instead of differences, just as one does when employing 
Lagrange's formula in the ordinary one-variable interpolation, and with the help of auxiliary 
tables such as those of Mr Herbert H. Edwards t very accurate results can be easily obtained. 
If however the intervals by which the given table of the function proceeds are different from 
those adopted by Mr Edwards, a large amount of arithmetical work is necessary. It is however 
worth while to recapitulate the method and increase the number of the independent variables 
from two to n. Assuming that we require to find «,.8( („) ; then by Lagrange's interpolation 
formula we have 
««.«... = *■<. 'im + ''i "i8( + »'2«2»<... etc., 
where ry, are numerical coefficients depending on the number of terms and the value of r. 
And similarly 
'W..— *oWO)'( .. + ''''l"lrt,.. + *2"2i-( ..) 
and so on ; hence 
«/•s^. = ■'o■'o^l... "iH»i..+ ''(i-'*(i<i ... Mtuii +etc. 
where S stands for the summation of all combinations of h, l\ I ... for all values from 0 upwards. 
The calculation of all the r^, Sj., terms is done directly from Lagrange's formula and is 
simple though laborious, and the remainder of the work is mechanical. When the function only 
depends on two variables the interpolation involves fairly heavy arithmetic if only three values 
are given to both h and k, while if there are three variables the work could only be completed in 
two or three hours, and it is therefore clear that if the ordinary one-variable interpolation 
formula could be used we should save a great deal of arithmetical work. 
The ideas underlying the two methods can be seen graphically. 
A 
i 1 
a:=0 
a- 1 
Fig. 2. 
In this figure a surface ACKG represents Ujc„, so that when :i:=0, y=% u„j has the value 
represented by the height A, and when ,r=l, ?/=!, u^,j has the value represented by E. Now if 
we require to find le , 4 we can approximate to the surface by the (1-t-Aj.) (H-A„) method or by 
the Lagrange coefficient method and hence find the particular height we require, or we can 
choose the positions of two or more values so that the straight line passing through them also 
passes through the point x=% ?/ = -4 and then interpolate by the one-variable formulae between 
the values of the function corresponding to the points on this straight line. In fig. 2 the line 
joining .r=0, y = 0 to .t-=l, y='2. passes through x=% y=-i and by interpolation between B and 
G we can therefore approximate to m.2,-4- The dotted lines show the process. 
* W. Palin Elderton in Biometrika, Vol. 11. Part 1. 
t Journal of the Institute of Actuaries, Vol. xl. pp. 289—293. 
