130 
Prohahle Errors of Frequency Constants 
In the actual construction of the tables fundamental values of jSy and /Sg were 
taken covering the usual range of these quantities in actual practice. The tables 
are, however, limited by certain considerations, which are not without suggestive- 
ness for theoretical frequency discussions. Along a certain line, indicated in our 
diagram, /Se becomes infinite. But this is statistically impossible. Hence, either 
we never get frequency distributions having such values of and fi^, or if we 
do the finite difference moment formula cannot approximate in such cases to the 
true state of affairs, and we must introduce an additional coefficient Cg into the 
denominator of the right-hand side of the fundamental differential equation. The 
discussion and classification of such curves are now in hand. On the diagram the 
areas where they are absolutely needful are marked Heterotypic. 
It is believed that the diagrams and tables now published will prove extremely 
useful to the bioraetrician. Having determined his /3i and /Sg, he will be able to 
see at a glance whether his frequency distribution may be safely treated by the 
usual types or is heterotypic. If it be one of the usual types, he will know at once 
how to classify it. Next an examination of Tables I., II., III. and IV., or such 
examination with a short interpolation, will give him with sufficient practical 
accuracy the probable errors of ySj, /3o, the modal divergence {d) and the skewness 
{Sh). Should other and more elaborate probable errors be required, they will be 
deducible from Tables I., II. and III., or by using Table VI., for all the usually 
desired constants depend upon the quantities therein tabulated. 
The two diagrams represent, (A), a practical working diagi'am covering the 
customary range of /3i and /3o and, (B), a diagram on a small scale showing the whole 
nature of the distribution of the type curves from the theoretical standpoint. 
The upper part of either diagram is bounded by the line 4/S2 — 3/9i = 0, because 
it has been shown that /S3 is of necessity > ^fii*'. The normal curve is represented 
by a point /3i = 0, /So = 3 marked G on the diagram. 
The transition Tyjjes III. and V. are given by values of /3i, ^« on the line 
W-2 - 3/3j -6 = 0, 
and on the cubic 
/3, (^, + 3)= = 4 (4/3, - 3A) (2A - 3/3, - G), 
respectively. 
Type II. is a particular case of Type I. for /3i = 0, and Type VII. a particular 
case of Type IV. for = 0 ; they are thus represented by portions of the vertical 
running through the " normal point " G. While the upper boundary of the diagram 
lies in the Z7-curve part of Type I., to be discussed below, the lower boundary is 
fixed by the line S/Sa — 15/Si — 36 = 0 along which the finite difference moment 
formula first fails. This matter needs a little consideration. 
* See Pearson : Phil. Trans. Vol. 186, A, pp. 343—414. 
