142 
MR. W. F. SHEPPARD OH THE APPLICATIOH OF THE THEORY OF 
by means of this table we can divide the base into 40,000 areas, each representing 
1/40,000 of the whole volume. To simplify the counting of the areas, every tenth 
line should be drawn in ink, the others being in pencil; a dot should be placed in 
each area, and the pencil lines should then be erased. There will thus be 400 larger 
areas, each containing 100 dots. It will be found convenient to replace the circle 
shown in fig. 10 by a larger graduated circle ; if the radius of this circle is p, and if 
Ox cuts the circumference of the circle in F, the line at right angles to Ox at a 
distance x from 0 will cut the circumference in points at an angular distance cos~^ x/p 
from F. 
The lines Ox, Oy, &c., may be shown on tracing-paper, instead of on the figure 
itself; and the paper may then be turned round O into two or three difterent posi¬ 
tions, so as to minimise inaccuracies of counting. Or the figure may be copied on to 
a glass plate, and the lines Ox, Oy, &c., drawn on ordinary paper. 
(ii.) The solid is a solid of revolution, and therefore can be divuded into mm equal 
portions by a set of m planes through the central ordinate at successive angular 
distances ^irlm, and a set of concentric cylinders enclosing portions Ijm', ‘lira, 
. . . {m' -- l)jm' of the whole volume. Let the rth cylinder cut a central section in 
the ordinate MP. Then, if OH is the central ordinate, r/m' = (OH — MP)/OH 
(§§ 5, 11). Hence the radii of the successive cylinders are the abscissse of the standard 
curve corresponding to ordinates whose ratios to the central ordinate are respectively 
[m' — l)lm', {in — 2)1111, . . . Ijin. Thus for m' = 100 the values are given by 
Table H. (p. 155). 
This method of division of the base-]3lane is not so convenient as the method 
explained in (i.), but it may be used for testing the accuracy of a figure constructed 
according to that method. If on such a figure we draw" circles with the radii given 
by Table IL, each of the rings so formed should contain one-hundredth of the total 
number of dots in the figure. Or, if we draw circles with radii •05, ’10, '15,. . ., the 
numbers in the successive rings should be proportional to the differences shown in the 
fourth column of Table I. 
(2.) A more accurate method can be adopted wdren the values of X and X', and 
also those of Y and Y', have been chosen so as to correspond to particular class- 
indices. Let these be a, a, /3, and respectHely, and let the corresponding 
abscissae of the standard normal figure be x, x , y, and y'. Thus (X — Li)/a = x, 
(X' - Li)/« = x', (Y - Mi)/6 = y, (Y' - Mi)/6 = y. Now if, by the method of 
§11, we construct a figure representing the division of the standard solid by parallel 
vertical planes at distances x and x from OH, and also a corresponding figure for 
distances y and y', the bases of the twm figures being in the same straight line, and 
the distance between corresponding extremities being equal to I)/27r of either base, 
the area formed by the two pairs of curves wall give the proportion of individuals for 
wdiich L lies betw^een X and X', and Y^ between Y and Y'. The most important case 
is that in which the class-indices for each distribution separately correspond to the 
