MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 375 
I ps — 16% va 3 
II = 001 OY = 2505 
ela p= ‘265 Y = 363 
IV p= 1021 a ‘7676 
vi D= 1 oh = 6) 
VI = 6°5625 Oe 4°3125 
VII p = 1890 y’ == 1700 
In the diagrams vertical and horizontal scales (y) and a) have been chosen so as 
to illustrate best the changes of shape in the curve. The general correspondence of 
this series with actual types of frequency curve, as indicated in Plate 7, fig. 1, will at 
once strike the reader. 
The mean, the median, and the mode or maximum-ordinate are marked by bd, cc, and 
ac, respectively, and as soon as the curves were drawn, a remarkable relation manifested 
itself between the position of these three quantities : the median, so long as p was 
positive, was seen to be about one-third from the mean towards the maximum. 
For p negative and between 0 and — 1, this relation was not true. The distance 
between the maximum-ordinate and the mean is, if the equation to the curve be 
y=yure™, 
equal to 1/y. Now the maximum cannot be accurately determined from observation, 
but a fair approximation can be made to the median. Hence the constant y could, if 
the above graphical relation were shown to be always true, be determined approxi- 
mately as the inverse of thrice the distance between median and mean. 
Now distance of mean from origin = (p + 1)/y, 
and is maximum eee—1 cys 
Hence, supposing distance of median =(p-+c)/y, we should expect to find 
¢ = 2/3 about. 
Equating the integral which gives the area up to the median to half the total 
area, we have 
Yo fie ere dx = ty | DL eCm dot, 
Ean 0 
y 
ao 
or, e 
oOo ie) 
| Came 4] ZiCmaOes 
pte 0 
This is the equation for c. Unable to solve it generally I gave p a series of integer 
values and found in all cases ¢ nearly ‘67. Its value, however, decreased as p 
