ccxxxii Tables for Statisticians and Biometricians [XL VI I XL VI 1 1 



right. The "exact" areas were found by quadrature, using Weddle's formula, two 

 figures being finally dropped. The ordinates were computed to eight figures and 

 thirty ordinates were used. We think the values are certainly exact to seven 

 figures. 



Formula A is Wishart's "Generalised Schlomilch Formula" (equations (viii) 

 and (xii)) and Formula B is the Expansion in Incomplete Normal Moment 

 Functions (equation (xiii)). See our pp. ccxxvi and ccxxviii. Formula B was "cor- 

 rected" as suggested by Wishart (Biometrika, Vol. xix. p. 26) by taking double 

 the last term to represent the last term and the remainder of the series in the case 

 of both the B and C series. 



Neither Formula A nor Formula B nor the two combined enable us to find the 

 area from either terminal to the mode, although this is needful if we require 

 the incomplete B-function, i.e. the probability integral of a Type I curve in the 

 ordinary sense. Accordingly, under Formula B for comparison with the "exact" 

 value, we can only use the ratio to the total area of the area from the mode up 

 to the ordinate corresponding to a given x (or u). 



The extent of the present table for a single Incomplete B-function indicates 

 that Formula A cannot be trusted to give even five-figure accuracy at 2'0 to 2'5 

 times the standard deviation from the mode; it becomes accurate at 3'0 times the 

 standard deviation on the shorter range side of the mode, but is not accurate to 

 the seventh figure till about 3'5 times the standard deviation from the mode on 

 the longer range side. 



Turning to Formula B, we see that precisely as in the case of the expansion in 

 a tetrachoric series, the degree of accuracy. is sinuous*, the error being sometimes 

 positive and sometimes negative. Thus there are values of u for which the error 

 is zero or very small, and if we happened to alight on one of these we might 

 imagine the expansion a good one. We see, however, that it is not generally 

 reliable beyond the distance of the standard deviation from the modef. Ac- 

 cordingly Formulae A and B leave a gap of roughly one to three times the standard 

 deviation where neither can invariably be safely applied; thus together they cannot 

 give the total area on either side of the mode. They serve only to find areas for 

 a round the mode, or to find tail areas in excess of + So- from the mode. 



We are accordingly within the above range (supposing the I and ra lie outside 

 the values of the Tables of the Incomplete B-function) reduced to using either (i) a 

 quadrature method preferably Weddle's, or (ii) the method of continued fractions. 



(i) We will indicate here first the process of obtaining an incomplete B-function 

 by Weddle's quadrature formula. 



* The two methods are closely related: see J. Henderson, Biometrika, Vol. xiv. pp. 157 158. 



t This conclusion was reached by the present Editor when in 1908 he attempted to apply the tables 

 of the Incomplete Normal Moment Function to the computing of Incomplete T- and B-functions. Its 

 non-recognition renders nugatory a good deal of Laplace's work in the Theorie analytique des pro- 

 babilites. Cf. Biometrika, Vol. vi. p. 68. 



