Miscdlanea 105 



II. Interpolation by Finite Difierences. {Tino Independent Variables.) 



By W. I'A[JN elderton. 



Let it lio reciuired tu lind «,,,. in terms of «qo, m^^j, ... «j.j ... m_ju ..., a^._^..., wliere ^ 

 and r luc both <1. 



«p„ = (l+Al)""0 :r = (l+A,)''(l+A,)'-M„,„ 



= 1 1 + (7'Ai + J-Aj) + — (^)(=) A,2 + 2pr A, Ao + )-(-lA/) 



+ ^,(/.P)Aj:' + 3/.l%A/^A, + 3/>rC^)A,A/ + rP)A,/)...JM„,„ (1), 



where /)(")=/? Q>- 1) ... (jo + ?i - 1) and A, and A.^ rei)rcsent opcration.s with resiiect to ^; and r 

 respectivcly. 



If we \i.se the expre.ssion {1 +(/iAi + rA2)j ?/„,„ for u,,.,. we reqnire only 3 value.s of tlie fiinction, 

 while if we take in the next term in round brackets we require 6 values, but the objeotion 

 to the t'ormuUe .seenis to he that the Viihies of the function which we use are not necessarily the 

 nearest vahies to Up.,. For practical purposes it would be better to have the expression in 

 terms of the function rather than in terms of the dift'erences, as the calculation of difterences 

 running in three directions is troublcsome and tlie work i.s likely to contain sHps*. 



The following scheme shews the form of the problem and gives an idea of which are the best 

 functions to use for interpolation ; 



•»l:-l «1:0 Mi:i «1,2 



«2.-1 «2:0 «2:1 "2:2- 



NOW «,,-0 = «0:0+.P(«1.0-«0:0). «P:l = "o:l+;'(«l:l-«0:l) 



and interpolating between these vakies we get (when s=\ - r and q=\ - p) 



«p-r = <i««0:0 + ?''«0:l+i'*'"l:0+/"'«l 1+ C^)- 



In most cases occurring in practice the coefficients in (2) can be calculated at sight and the 

 labour of the wliole Interpolation is very small. The coefficients can easily lie remenibered by 

 considering the distances of the requircd function from a given value and bearing in iniiid that 

 the nejirer the position of the given vahie to that of the required vahie the larger its coefficient. 



For some purposes (2) will not Ije sufficiently accurate and we must seek for a similar 

 formula involving inore terms. 



For this purpose Lagrange's Interpolation formula can be u.sed and would be the only one 

 applicable when the intervals between the given vahies of the function are not all equal. We 

 should first find w,, „, «,,,(,... and then «„^^ by independent interpolations or eise by working out 

 coefficients as in formula (2). If we consider Lagrange's formula, viz. 

 _ (/>-6)(jj-o)... (p-a){p-c)... 



"""-{a-b) (a-c) ... "^ {b -a) [b-c) ... ''^- 



[* The biometrician has ri^peatedly to use tablos of double entrj' : e.g. in tlie eases of skew Variation 

 ■when using the G-integral (Brit. Axsoc. Repoil, 1899), in dealing with goodness of fit (Biimutiiku, 

 Vol. I. p. 155), iu finding the iufluence of natural selection on conelation {Phil. Trans. A, Vol. 200, 

 p. 64), etc., etc. Hence the iniijortance of a good iiiethod of interpolation. Ei>.] 



t See note by T. G. Ackland, Juunuil Iitstiliilf of Actiiurii-.i, Vol. 32, p. 2815. 

 Biometrika ii 14 



