ERROR TO CASES OF NORMAL DISTRIBUTION AND CORRELATION. 141 
Below Lp 
Above Lj. 
Below Ml . 
P 
R 
Above LIi . 
R 
P 
the divergence is equal to ^ ^ tt. 
§ 28. Calculation of Table of Double Classif cation. —In the base-plane drav/ Ox, 
Oy at right angles to OX, OY, and therefore including an angle D. In Ox take 
ON = (X — Lj)/a, ON' = (X' — L,)/a ; and in Oy take 0?i = (Y — Mfb, 
On = (Y' — Mi)/6. Through these points draw vertical planes at right angles to 
Ox and Oy res^rectively ; then (§ 25) the volume of the portion of the standard solid 
included between these four planes represents the proportion of individuals for whicli 
L lies between X and X' and M between Y and Y'. 
The calculation of this volume requires the use of the integral calcidus. For a 
rough calculation we may use either of two methods. 
(1.) The planes by which the volume is bounded will meet the base-plane in lines 
forming a parallelogram, two of the sides of the parallelogram being at right angles 
to Ox, at distances (X — Li)/a and (X' — Li)/« from O, and the other two at right 
angles to Oy, at distances (Y — Mi)/5 and (Y' — Mi)/& from 0. Now suppose that 
the base-plane is divided up into very small areas such that the portions of the solid 
lying above these areas are all equal. Then the ratio of the number of these areas 
which lie inside the parallelogram to the total number will be the proportion of 
individuals for which L lies between X and X', and M between Y and Y'. For 
effecting this division of the base-plane into small areas we can use either of the two 
characteristic properties of the normal solid. 
(i.) The solid is a projective solid. Hence if we find the values of x corresponding 
to a = i 1/w, a = d: 2/7 ?i, , . . a = [ni — l)/m, and if we take the corresponding 
points on each of two rectangular axes £'0^, rj'Orj in the base-plane, and draw 
lines through these points parallel to y'Or) and to ^'0^ respectively, the two sets 
of lines will divide the base into Am~ areas, corresponding to the division of the solid 
into 4equal portions. Fig. 10 shows the arrangement of these lines for m = 50 ; 
thus the figure contains 10,000 rectangles (one or two of the sides of some of them 
being at infinity), and each rectangle represents 1/10,000 of the whole volume of the 
solid. The centre 0 of the figure is shown by a small circle. The larger circle is 
introduced to show the scale ; its radius is the semi-parameter of the solid, and is 
therefore the unit for measuring the distances (X — L)/a., &c. 
The values of x corresponding to m = 100 are given in Table VI. (p. 167); so that 
* This formula obviously applies in any case in which the correlation-solid is a solid of revolution. 
