106 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
rest of the population. It is clear that this population of giants is an unstable 
population, he., a very small disturbance would largely change its centre. That it 
accounts for and covers the dwarf and two giant anomalies is clear, and the resultant- 
curve, based on the addition of the two components, is a fairly close approach to the 
observation-curve—far closer indeed than that provided by the first solution, and a 
great advance on the normal-curve C, resulting from the observations as a whole (see 
fig. 5). I am inclined, accordingly, to suspect that the family of Prawns was not 
homogeneous, but contained between 2 and o per cent, of a giant population with a 
large standard deviation. Possibly the theory of correlations may settle whether this 
is the real state of the case, or whether the anomalies referred to ought to be rejected 
and a new investigation made to dissect the asymmetrical curve for the carapaces 
when the outlying parts, which control the nonic at present, are removed. 
The investigation of this case, however, with all the observations included, shows 
the great variety of solutions which may be suggested by the dissection of various 
anomalous and asymmetrical frequency-curves. 
Table I, — Powers of the Natural Numbers, 
Powers. 
First. 
Second. 
Third. 
Fourth. 
Fifth. 
Sixth. 
I 
1 
1 
1 
1 
1 
2 
4 
8 
16 
32 
64 
3 
9 
27 
81 
243 
729 
4 
16 
64 
256 
1,024 
4,096 
.5 
25 
125 
625 
3,125 
15,625 
6 
36 
216 
1,296 
7,776 
46,656 
7 
49 
343 
2,401 
16,807 
117,649 
8 
64 
512 
4,096 
32,768 
262,144 
9 
81 
729 
6,561 
59,049 
531,441 
10 
100 
1,000 
10,000 
100,000 
1 .000,000 
II 
121 
1,331 
14,641 
161,051 
1,771,561 
12 
144 
1,728 
20,736 
248,832 
2,985,984 
13 
169 
2,197 
28,561 
371,293 
4,826,809 
14 
196 
2,744 
38,416 
537,824 
7,529,536 
15 
225 
3,375 
50,625 
759,375 
11,390,625 
16 
256 
4,096 
65,536 
1,048,576 
16,777,216 
17 
289 
4,913 
83,521 
1,419,857 
24,137,569 
18 
324 
5,832 
104,976 
1,889,568 
34,012.224 
19 
361 
6,859 
130,321 
2,476,099 
47,045,8sl 
20 
400 
8,000 
160,000 
3,200,000 
64,000,000 
21 
441 
9,261 
194,481 
4,084,101 
85,766,121 : 
22 
484 
10,648 
234,256 
5,153,632 
113,379,904 i 
23 
529 
12,167 
279,841 
6,436,343 
148,035,889 
24 
576 
13,824 
.331,776 
7,962,624 
191,102,976 
25 
625 
15,625 
390,625 
9,765,625 
244,140,625 i 
26 
676 
17,576 
456,976 
11,881,376 
308,915,776 
27 
729 
19,683 
531,441 
14,348,907 
387,420,489 
28 
784 
21,952 
614,656 
17,210,368 
481,890,304 ' 
29 
841 
24,389 
707,281 
20,511,149 
594,823,321 
30 
900 
27,000 
810,000 
24,300,000 
729,000,000 
