FOUNDATIONS OF THEORETICAL STATISTICS. 
345 
In Type IV. 
df cc 
»' + 2 
2 — v tan' 1 — 
e « 
we have written v, not as previously for the difference between m l and m 2 , for these 
quantities are now complex, and their difference is a pure imaginary, but for the differ¬ 
ence divided by \ 7 —1 ; v is then real and finite throughout Type IV., and it vanishes 
along the line NS, representing the symmetrical curves of Type VII. 
/ 
df oc[l + ^ 2 
\ <r ; 
from r = cc to r = — 1. 
The Pearsonian system of frequency curves has hitherto been represented by the 
diagram (13, p. 06), in which the co-ordinates are fa 1 and fa,. This is an unsymmetrical 
diagram which, since fa 1 is necessarily positive, places the symmetrical curves on a 
boundary, whereas they are the central types from which the unsymmetrical curves 
diverge on either hand ; further, neither of the limiting conditions of these curves can 
be shown on the ft diagram ; the limit of the U curves is left obscure,* and the other 
limits are either projected to infinity, or, what is still more troublesome, the line at 
infinity cuts across the diagram, as occurs along the line r — 3, for there fa., becomes 
infinite. This diagram thus excludes all curves of Types VII., IV., V., and VI., for which 
r < 3. 
In the fa diagram the condition v = constant yields a system of concurrent straight 
lines. The basis of the representation in fig. 2 lies in making these lines parallel and 
horizontal, so that the ordinate is a function of r only. We have chosen r — y— -, 
y 
and have represented the limiting types by the simplest geometrical forms, straight lines 
and parabolas, by taking 
i _ * , a _ (I + V + x~) (1 ~ U — xd 
y [ar + y) 
It might have been thought that use could have been made of the criterion, 
C> o 
_ _ ft i (^2+3)“ _ _ L _ fi!. 
Ka 4 (4/3 2 — (2fa 3 —Sfa 1 — 6) 4e’ 
by which Pearson distinguishes these curves ; but this criterion is only valid in the 
region treated by Pearson. For when r — 0, k 2 — 1, and we should have to place 
a variety of curves of Types VII., TV., V., and VI., all in Type V. in order to adhere to 
the criterion. 
This diagram gives, I believe, the simplest possible conspectus of the whole of the 
Pearsonian system of curves ; the inclusion of the curves beyond r = 3 becomes neces- 
* The true limit is the line /L = fai + l, along which the curves degenerate into simple dichotomies. 
