PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 101 
by tlie roots of this equation suggests very considerable divergences and peculiarities 
as likely to arise, when a considerable number of frequency-curves are dealt with. 
The discussion of the case of Crabs must not be taken as indicating that the 
incidents of this case will be generally true for other groups of biological measure¬ 
ments, until a very great variety of such groups of measurements have been 
mathematically analyzed. 
In order to thi’ow more light on the general question, I have added the following 
analysis for the case of Prawns, the measurements for which were kindly placed at 
my disposal by Mr. H. Thompson, who has been making elaborate measurements of 
1,000 specimens in the Zoological Laboratory of University College, London. 
PcdcBmon serratus .—Measurements in 998 ? specimens (adult) from penultimate 
to hindmost tooth on the carapace. 
^leasurements reduced 
to tLousandtlis of body 
length. 
Number of specimens. 
Measurements reduced 
to thousandths of body 
length. 
Numbei' of sjiecimens. 
27 
1 
49 
25 
28 
0 
50 
17 
29 
0 
51 
11 
30 
0 
52 
8 
31 
1 
53 
4 
32 
0 
54 
1 
33 
3 
55 
0 
34 
3 
56 
0 
35 
4 
57 
1 
36 
11 
58 
1 
37 
24 
59 
0 
38 
38 
60 
0 
39 
56 
61 
0 
40 
80 
62 
0 
41 
105 
63 
0 
42 
121 
64 
0 
43 
117 
65 
1 
44 
108 
66 
0 
45 
77 
67 
0 
46 
69 
68 
0 
47 
62 
69 
1 
48 
48 
The novel and somewhat remarkable feature in these results are the “ giants ” at 
65 and 69. To neglect these giants, as in some degree anomalous, would, no doubt 
be convenient, so far as the analysis is concerned, and would lead to a simpler reduction 
of the group. They have, however, been retained as among the data given to me, 
and their presence affords an interesting illustration of the various singularities which 
may arise in the solution of the fundamental nonic. 
(17.) The curve (see fig. 4) given by the observed numbers will be at once seen to 
