374 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
If we transfer the origin to the centroid-vertical we have 
2B 3 28/43? — 1 
2 fg" | fs / 
e —2pot/Kg 

Yi (1 ae 
where 
fOr (p+ Dyer" (p + UP 
NS Oa) Tp +1) 

It is interesting to note how this skew curve passes into the normal curve when 
pz is made vanishingly small, or p= o. 
By WALLIs’s theorem the limit to y, = #/4/27p3. 
It remains to find the limit of 
N\A Bo 1 xg \P Vp le/" as 
al: + e “Pat/Ks —— e 
\ 

2 p9?| pes au VJ (p+ 1) VA (He) 
= [ia + u) aril 
Now the limit of {(1 + u)e~"}" for u = 0 is easily found to be e ~?, hence 
y = a= *P/ (Om) 
the normal form. 
Returning to the value we have found for p, and eliminating p and y between yy, 
Ps, and pw, we find 
2 pho (Spa? — py) + Bp" = 0. 
This is the expression (see p. 8398) which must be positive in the case of limited 
range. It is zero also for the normal curve, because both 3u,” — pw, and ps vanish. 
Hence the more nearly the quantity 2u, (32° — 4) + 8p,” approaches to zero, the 
more nearly are we able to fit our statistics with a skew frequency-curve having 
a range limited in one direction only. 
(18 bis).—The skew frequency-curve of Type III. deserves especial notice. It is 
intermediate between those of Type I. and Type IV., and they differ very little from 
it in appearance. Hence, if the reader has once studied the various forms which 
Type III. can take as we alter its constants, he will grasp at once the forms taken by 
Types I. and IV., by simply considering the range doubly limited or doubly unlimited. 
To assist the process of realising Type III., Plate 9, fig. 5, has been constructed ; it 
contains seven sub-types of this species, varying from fig. 1., in which the curve is 
asymptotic to the maximum frequency-ordinate to fig. vil, which is practically 
identical with the normal curve. Taking y= y (1 + «/a)’e~’** for the equation 
to the curve, we have the following values for the constants p, and y’ :— 
