376 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
increased. I, therefore, assumed c to be really of the form ¢ = c, 4 c,/p, and deter- 
mining ¢, and ¢, by the method of least squares, found 
c = 6691 + -0094/p. 
Probably this is only the beginning of a rapidly converging series in inverse powers 
of p, but it would appear to suffice for most practical purposes. It is only true for 
p > 1 and does not explain why, when p is positive and fractional, c is still apparently 
near 3; thus its value for p = 0 has only risen to ‘6931. We have then the following 
fairly simple means of determining roughly the constants of a skew curve of this type: 
(1.) Find the mean and the median ; these gives y, approximately. 
(2.) Find p, for the mean; this gives p, since py, = (p + 1)/y’*. 
(3.) Knowing p, correct the value of y by using the above value for ¢, and so obtain 
a corrected p. 
(4.) Determine y, from the area. 
This method is not very laborious and may be of service in some cases.* It will, 
of course, fail for any curves in which p is negative, and must only be applied when 
the curve is known to be of Type III. If the beginning of the range is definitely 
known, we may save stage (2) above and find p from the distance of the mean from 
the start of the range. 
(19.) Generalised Probability Curve of Type IV. Range unlimited, but form skew. 
7] — Yo 4» —v tan—! (w/a) 
1 > = Gia 
Put « = a tan 0, hence 
NS Oyj Gos *0a 
ace 71/2 
op, = | yar" dx = Yo qeth | cose?” 24 sin”@ e7”? dé : 
a —n/2 
7/2 
= Oyen | cos’~"@ sin” 0 e~” dO, if r = 2m — 2, 
—n/? 
Ye" apa (e 
Se yh (Il 
Sen ee 
n/e 
1/2 
cos’ "*?@ sin™-"6e-" d@ — v| cos’~**!@ sin” '8e—” dé I 
—7/2 
a 
— poe {( sone 1) Op n—2 me rp aaa be, s 
provided 7 > n — 1. 
* The points of inflexion may also occasionally be found from the observations; they are at distances 
+E /»/- on either side of the maximum ordinate. 
