xiv Tables for Statisticians 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, namely we are 

 given u o (0) and have to determine from it. If we only go as far as second 

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

 generally be obvious without ambiguity. Usually it suffices to find 



and then determine from 



0=( ( , O (0)- ? , O )/A"o+^ (1 2 7 0,) AV^'«. (iv); 



or to find & = (h„(0) - «„)/£ (Ah„ + Am_,) 



and then 



6 - («. w - a.vi (a* + *o - f; HA ffXj (v) - 



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

 polation formula which for three values of u reduces to 



u o (0) = (l-0 1 )n v -^0(l-0)u^ l + ^0(l+0)u 1 (vi). 



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



u o (0) = (l - 0)(1 ~ i0)r, o + \0 (o - 0) Ul -l0(l - 0)(u_ i + u 2 ) ... (vii). 

 The resulting quadratics are respectively: 



0-- (J («, + m_,) - «„) + 1 \ («, - u_,) + U, - u, (5) = (vi) bis , 



and s ^ («, - «, + '/_, + m 2 ) + \ (5;*! - 5w — it_, - M,) + u, - m (0) = . . .(vii) bis . 



(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 : 



«*,!, = «So + a*A«M + Z/A'wo,o 



+ 4 {#(* - 1) AX,o + 2ay A A'w„,n + y(y-l) A%„) (viii>, 



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

 we consider u x<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, G, D, &c. the adjacent points at which the ordinates are known from the 

 table of double entry, only the points A, B, C, D, 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, C, D — 

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



