xiv Tables for Statistkkins and Biometricians [Interpolation 



It is very rarely indeed that we need go beyond second differences, often the 

 first will suffice. Not infrequently the inverse problem arises, namel}' we are 

 given Mo(^) and have to determine 6 from it. If we only go as far as second 

 differences, either (i) or (iii) gives us a quadratic to find 6 and the root will 

 generally be obvious without ambiguity. Usually it suffices to find 



and then determine from 



■d=(u,{0)~u,)jAu„+ —^. ^A-u„/Au, (iv); 



or to find 6' = (ii, (6) - n„)/A (tlu, + Ah_,) 



and then 



^H'U^)-»„)/MA., + AO-f^^^^|^) (V). 



Very often good results are readily obtained by applying Lagrange's inter- 

 polation formula which for three values of u reduces to 



u,ie) = {i-e')u,-W{i-e)u_,+^d{i + e)u, (vi). 



Or, we may use the mean of two such formulae and take 



Uo(e)={i-0){i-ie)n, + ie{5-d)u,-i6(i-e)iu_, + u.;) ... (vii). 



The resulting quadratics are respectivelj' : 



0' (H"i + w-i) - «o) + 61 H"i - "-i) + "u - ^'„ (^) = {v[)^'\ 



and 6^ I (»o - "i + u_, + »,) +01 {5ui - oii„ - u_, - lu) + u, - u, {6) = .. .(vii)'"^ 



(2) There are some tables in this book which are of double entry, e.g. those 

 for the Tetrachoric Functions and for the G (r, v) Integrals. The simplest solid 

 interpolation formula, using second differences, is : 



Ux,y= >io,« + a'A(/o_„ + yA''«o,o 



+ ^ {« {x - 1) Ahio, + 2.^■y A A'2<„, , + y(y-l) A'-«o, »} ( viii), 



where A denotes a difference with regard to .c, and A' with regard to y. But if 

 we consider u^^y to be the ordinate of a surface, and the figure, p. xv, to represent 

 the xy plane of such a surface, then it is clear that, if P be the point x, y, and 

 A, B, C, D, &c. the adjacent points at which the ordinates are known fi'om the 

 table of double entr}', only the points A, B, C, It, J, and N are used by the above 

 formula ; and of these points, not equal weight is given to the fundamental points 

 A, B, G, D, for G only appears in a second difference. If another point of 

 the fundamental square other than A be taken as origin, we get a divergent, 

 occasionally a widely divergent result. If we use only four points — A, B, G, D — 

 to determine the value of the function at P, then we might take the ordinate 



