448 
PROF. KARL PEARSON ON SKEW VARIATION. 
But if cr = 0, b must be finite or zero, and these both again throw us back on a 
concentrated frequency at x = 0. 
Accordingly, when ft x and ft 2 both become infinite, we deal with a concentrated 
frequency lump. But the ratio of ft 1 to ft 2 will depend on the manner in which we 
have reached this limiting case. 
For example, if we are dealing with the correlations in samples of two drawn from 
a population in which the correlation is p , the frequency consists of two lumps, but 
as p approaches unity, one lump shrivels up, ft x and /3 2 both become infinite, but their 
ratio is one of equality, i.e., we approach infinity along the line /3 2 —ft x — 1 = 0. 
When we take samples of three from a population of correlation p, the frequency 
curves are U-shaped, but as p approaches unity the frequency concentrates in one leg 
of the U, ft 1 and ft 2 both become indefinitely larger, but their ultimate ratio ft 2 /ft 1 
appears to equal f-. # The U-curve flattens down into an L-curve, of which the 
horizontal limb extends to infinity and becomes indefinitely thin, while the vertical 
limb contains all the frequency. 
(9) Scheme, of Skew Frequency Curves Represented as a Diagram. —We are 
now able to considerably enlarge our diagrammatic representation of frequency 
curves. (See Diagram, Plate 1.) 
Every distribution is represented by its characteristic co-ordinates ft l and ft 2 , which 
must be positive, and therefore we need only deal with the positive ft u ft 2 quadrant. 
No frequency distribution at all can lie above the line ft 2 — /3 X — 1 = 0 ; this restriction 
removes more than half the positive quadrant. No frequency distribution can be 
adequately represented by one of the present system of skew curves, if it falls below 
the line 8ft 2 — 15/^ — 36 = 0. The area below this line is therefore termed heterotypic. 
Heterotypic distributions are to say the least of it very rare, if they be not extremely 
improbable. We have seen that there is some reason to suppose that bimodal 
distributions would give rise to such heterotypic distributions, but with our present 
views as to frequency such distributions when they do not arise from the mere 
anomalies of random sampling are classed as heterogeneous, and supposed to be due 
to mixtures. 
Having thus limited our area at top and bottom we proceed to consider the various 
possibilities that arise. 
The ft.,- axis, where ft x — 0, is the axis of symmetrical frequency distributions. 
Possibilities begin at the B-line or the point ft 2 — 1, or we have two equal concen¬ 
trated frequency blocks at any arbitrary distance b. This is the case of two 
alternative values, either of which is equally probable. For example, heads or tails 
in the repeated tossings of a single coin, or positive or negative perfect correlation 
in samples of two taken from a population of individuals bearing two uncorrelated 
* I use the word “appears ” advisedly, because the ratio has been obtained by determining the value 
of ft 2 /fti for high numerical value of (>. The actual ratio for p = 1 depends upon approaching a limit 
in rather complicated elliptic integral expressions, which I have not yet accomplished. 
