1899] VARIATION-STATISTICS [IN ZOOLOGY 329 
but also in those of species belonging to different genera, even to 
different families. This fact does not seem to me to have been 
sufficiently regarded hitherto; the explanation of it is, I suppose, the 
constancy of the physiological capacity of a given organ for reacting to 
the individual causes of variation (to be considered afterwards) with 
respect to a given character. Some authors, however, seem to assume 
a more or less constant relation between the height of the average 
and that of the index of variability of a character. 
Average value and index of variability of a numerical character 
are the first data necessary to the description of its variation. Both 
ought always to be determined; but they only give an approximate 
idea of the variation of the character. Its complete description 
requires the determination of the curve which rules the slope of its 
polygon of variation, or in other words, on which the corner points of 
the polygon are situated. To find this curve, we must find the mathe- 
matical relations between the variants or their deviations from the 
average value on one hand and their frequencies on the other. 
There is a striking likeness, even at the first glance, between the 
polygons of variation and binomial polygons. We get the latter by 
graphically representing the series of summation which arise by 
developing binomial terms, as (p+q). As a matter of fact, both are 
closely related. In numerical characters we find variants deviating 
from the average value in positive and in negative directions. Since 
all processes in nature depend upon causes, we are obliged to assume 
causes of variation with either positive or negative effects, of which 
causes neither the number nor the intensity of effect is known. These 
causes must be different from those which determine the average 
character of the form-unit, and at the same time must be weaker in 
effect than the latter. Now each individual has its own fate, which 
word includes the total sum of enormously numerous and minute 
factors acting on it in the most diverse combinations, which naturally 
cannot be identical either for all individuals of the form-unit or in 
every moment of the existence of the single individual. Thus we get 
the conception of an enormous number of elementary causes of variation, 
which may be regarded as equally effective so far as their small power 
goes. Of these one set can effect positive deviations, the other negative 
deviations from the average values of the different characters, all being 
able to act on each individual of the form-unit, though as a matter of 
fact they do not all act on it. The active set of causes is in each case 
any random combination of positively and negatively acting causes, 
and each of these combinations has a higher or lower degree of 
probability, according to which its effect is more or less frequent in the 
total population of individuals. The sum total of positively acting 
causes may be equal in number to that of negatively acting ones, or be 
different from it. 
Starting from such assumptions, mathematicians have investigated 
