358 MR. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
of X’.,, as compared with @ is unjustifiable. Thus, the maximum of a fever 
mortality curve cannot be an infinite distance from birth, which limits the curve in 
one direction, nor an age-at-marriage curve have a maximum frequency infinitely 
distant from the age of puberty, nor a frequency of interest curve separate its 
maximum, between 3 or 4 per cent., by an infinite distance from 0 per cent. It is 
clear, therefore, that if such frequency curves as those referred to are to be treated 
as chance distributions at all, it would be idle to compare them to the limit of a 
symmetrical binomial. We are really quite ignorant as to the nature of the contri- 
butory ‘‘ causes” in biological, physical, or economic frequency curves. The continuity 
of such frequency curves may depend upon other features than the magnitude of n. 
If I toss twenty coins, a discrete series of 0, 1, 2, 3,... 20, heads is the only possible 
‘contributory cause’ 
¢ ? 
range of results. Each individual coin, here representing a 
can only give head or tail, and so many whole coins must give head, so many tail. 
If I want to make any ratio of head to tail, I have to take an indefinitely great 
number of coins, for each ‘ contributory cause” must give a unit to the total. But 
it may possibly be that continuity in biological or physical frequency curves may 
arise from a limited number of “contributory causes” with a power of fractionizing 
the result. We cannot conceive on the tossing of 20 coins that 13°5 will give heads 
and 6°5 will give tails, we are obliged to deal with 200 coins, 135 giving heads and 
65 tails. Yet the two things are not identical. The former corresponds to a value 
intermediate between two ordinates of (4 + 4)*°, and the latter to a definite ordinate 
of (+4). So long as we remain in ignorance of the nature and number of 
‘contributory causes” in physics and biology, so long as we do find markedly skew 
distributions, it seems to me that we must seek more general results than flow 
from the assumption that p = g and i = «. The form of curve given in § 8 above is 
suggested as a possible form for skew frequency curves. Its justification lies 
essentially, like that of the normal curve, in its capacity to express statistical 
observations. : 
(10.) But it must be noted that the generalised probability curve in § 8, although 
it contains the normal curve as a special case, is not sufficiently general. It is 
limited in one direction, indefinitely extended in the other. This limitation at one 
end only, corresponds theoretically to many cases in economics, physics, and biology. 
But there are a great variety of cases in which there is theoretical limitation at both 
Cc 

D 
ends ; that is to say, there is a limited range of possible deviations. For example, 
let a trapezium, ABCD, of white paper be pasted on a cylinder of black surface with 
