H. E. Crampton 
U7 
occasion to recur to this point below. But at any rate elimination of the second 
period need take no account of this possibility. The antennal characters remain 
unchanged after death. 
It will be seen that the " length " of the antenna is only a convenient measure 
approximating to the real axial length, which dimension it is almost impossible to 
measure directly. From this it follows that the determination of the antennal 
proportions is likewise only a convenient approximation, and the value of these 
characters needs to be further considered below. 
Following the usual procedui'e, the determinations of every character in each 
sex were classified, and the frequency curve or pol3'gon was plotted. It will be seen 
that each sex will have for every character five groups: all pupae, surviving pupae, 
dead pupae, perfect survivors, and the survivors which fail to metamorphose 
perfectly, which last we may term the imperfect survivors. The characters of the 
whole group need not concern us at this juncture. Now the comparison between 
the surviving and dead pupae, or between the two sub-groups of survivors, may be 
based upon the following constants : (1) the range of variation, (2) the mode, or 
value of greatest frequency, (3) the average value, or mean, (4) the standard 
deviation, or the index of variability, a constant found by squaring all the 
deviations from the mean, adding, extracting the square root, and dividing by the 
number of cases, and (5) the coefiicient of variation, a constant of use in some 
connections, found by dividing the standard deviation by the mean, and multiplying 
the quotient by 100 to obtain a convenient whole number. Naturally the data at 
hand permit of much further mathematical analysis, particularly as regards the 
phenomena of correlation. For the present however attention may be directed 
only to the fundamental question as to the relation between variation and 
elimination and the comparison, for the sake of simplicity, may be restricted to 
the comparable determinations of the mea)i or average value of any given 
characters, and of the indices of variation. 
Before passing to the actual facts, it must be pointed out that statistical 
treatment of the problem under consideration rests upon the assumption that 
when differences are found between two corresponding constants, belonging to 
two different groups, such differences are significant only when they considerably 
exceed the probable errors of the differences. For example, the bust length of 
perfect surviving pupae has a certain value for which we may determine the 
probable error, while the bust length of the comparable imperfect survivors has 
its mean value with a probable error. If the difference between the two mean 
values in question should exceed the probable error of such a difference, the 
chances that the difference signifies selection will be 68 to 32 ; if the difference 
exceeds twice the error of the difference the chances of significance are 95 to 5 ; 
while if the difference is in excess of three times the probable error the chances 
of significance are 975 in 1000. Therefore in the following account, a difference 
between two comparable determinations which lies between the values of le and 2e 
will be regarded as indicating that selection is here possible, when the difference 
