360 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
is not so, and under any circumstances the limited curve may actually give information 
as to the possible range—the “ limits of stability ”—which is itself of great value. 
We have, then, reached this point: that to deal effectively with statistics we 
require generalised probability curves which include the factors of skewness and range. 
The generalised curve we have already reached, possesses skewness, but its range 
is limited in one direction only. 
Accordingly, we require the following types of frequency curves :— 
Type I.—ULimited range in both directions, and skewness. 
Type II.—Limited range and symmetry. 
Type ITI.—Limited range in one direction only and skewness. 
Type IV.—Unlimited range in both directions and skewness. 
Type V.—Unlimited range in both directions and symmetry. 
Type V. is the normal curve; Type IV., with slight skewness, has been dealt 
with by Porsson in the form of an approximative series.* Type III. has been given 
above, it was first published by me without discussion in ‘ Roy. Soc. Proe.,’ vol, 54, 
p. 331. 
We can now turn to the general problem. 
(11.) A very simple example will illustrate how a frequency curve, with limited 
range and skewness, may be considered to arise. ‘Take n balls in a bag, of 
which pn are black, and qv are white, and let 7 balls be drawn and the number 
of black be recorded. Ifr>pn, the range of black balls will lie between o and pn; 
the resulting frequency polygon will be skew and limited in range. This polygon, 
which is given by a hypergeometrical series, leads us to generalised probability 
curves, in the same manner as the symmetrical and skew binomials lead us 
to special cases of such curves. If we consider our balls to become fine shot, or 
ultimately sand, and suppose each individual grain to have an equal chance of being 
drawn, we obtain a continuous curve.t It is not, however, impossible that, could we 
measure with sufficient accuracy, many physical as well as biological statistics might 
be found to proceed by units, much as in certain types of economic statistics we are 
not troubled with fractions of a penny. For this reason we shall keep our results 
in the most general form, and obtain a curve approximating to the hypergeo- 
metrical series referred to without any assumptions as to the relative magnitude of 
the quantities involved. 
We easily obtain for the series giving the chances of r, 7 — 1,7 —2... 0, black 
balls being drawn out of a bag containing pn, black, and qn, white, the expression 
* “Sur la Probabilité des Jugements,” chapter 3. 
+p pints of red sand and gq pints of white sand are put into a vessel, and 7 pints are withdrawn. We 
have if 7 >, a perfectly continuous frequency curve for red sand withdrawn ranging between o and p 
pints. We are here supposing no “ perfect mixture” of the two kinds of sand, but theoretical equality 
of chances for each grain, 
