MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 373 
(18.) Generalised Probability Curve of the Type III. Range limited in one 
direction only. 
y= (1 + a/ayre 
In this case we have no need to determine the value of ,, and the analysis is much 
simplified by the replacement of the B function by a single I function. 
Take z = y(a-+ x) and write ya = p, we have 


SE per 
ae 
Further, « = —a,z=0,x=0.z=0. Thus we find 
= ip n pean Ye? [- +n p—2 
Opts — [y (x + a)” da = eee geese dz. 
Hence 
_ youe? p ANG ae ae 1) 
aaa I(GD sb Oy chip A=) 
whence 
en Distal 7 (sea ae 2) 
A= via (hy = ahaa ey sey 


pu (Pick Diet 2) (pin) ,__ (p+ 1)(p + 2) (p+ 3) (p + 4) 
Bs = x? ? hy = yy! ; 
Or, transposing to the centroid-vertical, we have 
Dee 2(p +1) 
Sent ee = ee wee 


es (etn) (ak 3). 
yf ; y' 
The first two results give us at once 
Y = 2p/b3, P= 4py?/ps” —- 1, 
whence 
“a po 

eels eG soe poe eee 
Yo pg) 2p? and: oar a, a0 (Oe Ds 
This completes the solution of the problem, which is seen to require only the 
determination of p, and ps, 
Remarks.—The distance d of the centroid-vertical from the axis of y or maximum 
ordinate d, is given by 
Thus 
skewness = d/./ pz = $p3/Po"”. 
