xiv Tables for Statisticians a ml Hinni, triciaii* | INTKHI-OLATION 



It is very rarely indei-d that we need go beyond second differences, often the 

 tiivt will suffice. Not infrequently the inverse problem aris.-s. namely w<- are 

 given M(0) and have to determine 6 from it. If we only go '^ ^ ar second 

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

 be obvious without ambiguity. Usually it suthYrs to find 



.ind t.hi-n determine ft from 



f) = (M . (ft) - .)/A., + ~ Ai/../A .................... (iv) ; 



or to find fP = ((#) - )/ (Ai/<, + AH_,) 



and then 



0* A 



.(v). 



Very often good results an- readily obtained by applying fvigrange's inter- 

 polation formula which for three value* of n reduces to 



0)M, ............... (Vi). 



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



.(0) = (l-0)(l-{0) +i0(5-0) 1 -i0(l-0)(H_ 1 + a ) ... (vii). 

 The resulting quadratics arc respectively : 



* (i (,+-,)- Wo) + tfi(M,--,) + .-W.(fl)-0 ......... (Vi) b * ( 



and ^ i ( - M, + (_, + ) + ^ \ (5! - 5 - _, - .,) + ?/ - u (6) = . . .(vii) 1 * 1 . 



(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 : 



l)A'Xo) ...... (viii), 



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

 we consider u XtV 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 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 C 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 



