370 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
Now it is theoretically impossible to fit a normal curve (uw, = 34”) to a frequency 
distribution for which py > 3p”. It is, however, possible to fit this generalised curve 
of Type L, although p* be >3y,”, provided there is sufficient skewness to render 
Sig” > 2p (fy — Bpy’). 
Hence the first stage in determining the type of curve suitable for a given set of 
observations is to ascertain the value of 
2p (Bptg” — py) + 3y45°. 
If this expression be positive, we see that a limited range of variation is a possibility. 
Passing from range to skewness we remark that the distance d between the centroid 
vertical and the maximum ordinate 
= 4, — p= a, — bm’, /(m’', +m’), 
__ Gm’, — am’, 
m, + m’, 
b (m, — mg) 
(a, + Mz) (mM, + Mz + 2) : 

Now it might seem that d/b would form a good measure of skewness, and it would 
be so if all curves had a limited range. But, as they have not, it seems to me 
better to take as the measure of skewness the ratio of the distance between the 
maximum ordinate and the centroid to the length of the swing radius of the curve 
about the centroid vertical, z.e., the quantity d/,/py. 
In our case we have accordingly, 
Mh — Mg a/ (mm +m +3 * 
skewness = ———— 2 1 
My + Mg a + 1)(m, + 1), 
~ r+2 
=tVB, 9 
{PZ 


in our previous notation.* 
Thus range and skewness are determined in Type I. 
(15.) A very considerable simplification of the above analysis arises when the range 
is given by the conditions of the problem itself, ¢.g., guessing between two given tints. 
In this we only require the moments p’, and p’, about one end of the range, and the 
solution becomes as easy as in the case of fitting a normal curve. 
Since b, pw’, and p’, are known, let 
Y= H3/b and yo = p'p/(w4). 
* The points of inflexion of the curve are at distances + Va a,/(m, + mz — 1) on either side of the 
maximum ordinate. 
