]\rATHEMATTGAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 295 
0/4, 1/4, 2/4, 3/4, 4/4, 
0/5, 1/5, 2/5, 3/5, 4/5, 5/5, 
0/6, 1/6, 2/6, 3/6, 4/6, 5/6, 6/6, 
0/26, 1/26, 2/26, .... 25/26, 26/26, 
were equally likely, how would the frequency depend on the grouping 
Taking 26 coverings as the probable maximum—it actually occurs—we have for 
the total number of fecundities given above ; 5d-6 + 7-l-----r27 = 368 separate 
fecundities. Let us see how they woidd he divided in one or two cases. 
Case (i.) Let the elements be based on 1/8, or be 0-1/16, 1/16-3/16, 3/16-5/16, 
5/16-7/16, 7/16-9/16, 9/16-11/16, 11/16-13/16, 13/16-15/16, 15/16-1. 
The half-groups at the ends are taken so that zero and perfect fecundity should 
really be plotted at the middle of a 1/8 element. We find, adding up the numbers 
of the above fecundities which fall into the nine groups, the folloAving frequencies :— 
33-5, 42, 43-.5, 44, 42, 44, 43-5, 42, 33-5. 
There is thus a somewhat deficient frequency in the terminal groups, and this 
would probably to some extent bias the distribution. 
Case (ii.) Let the elements be based on 1/15, or be 
0-1/30, 1/30-3/30, 3/30-5/30, . . . 25/30-27/30, 27/30-29/30, 29/30-1. 
We liave the following distribution : 
23, 
22, 23, 23-5, 22*5, 23-5, 23 5, 22, 22, 23-5, 23-5, 22-5, 23-5, 23, 22, 23. 
The bias here is only slight and the distribution is on the whole very satisfactory. 
Case (iii.) .Let the elements be based on 1/20, or be 
0-1/40, 1/40-3/40, 3/40-5/40, . . . 35/40-37/40, 37/40-39/40, 39/40-1. 
We find for the groups : 
23, 13, 17-5, 17-5, 17, 18, 17, 18, 17-5, 17, 18, 17, 17-5, 18, 17, 18, 17, 17-5, 17-5, 13, 23. 
Here the terminal groups have too great a frequency, and the adjacent groups too 
little. It is clear that the division into 1/15 elements is better than those of 1/8 of 
1/20, so far as these results go. But unfortunately the different coverings do not 
occur in anything like tlie same proportions. Their exact frequencies could only be 
found d posteriori, and I was desirous of having some idea of grouping before start- 
'* Such problems are really not infrequent in statistical investigation.s, and seem to be of some 
interest for the theory of fractional numbers. Mr. Fxlon worked out for me the details of the cases 
given below. 
