102 PROF. K. PEARSON ON TEE MATHEMATICAL THEORY OF EVOLUTION. 
be distinctly asymmetrical. Adopting the carapace length 31 as the origin of 
coordinates, and using the same notation as before, we have the following results :— 
= d{= q)= lG-191,382,8 
P2= 270-277,555 
= 4,963-876,753,5 
= 94,386-734,469 
/x's = 1,920,725-520,040 
/Xl = 0 
11.2 = 14-116,678,13 
P3 = 33-424,02673 
1,288-640,094,26 
p,5 = 16,752-563,9961 
- 
2072-394,903 
X. = - 36,102-605,1706. 
The standard-deviation of the group as a whole is given by cr = \/po, or 
o- = 3-7572. 
The mean errort obtained from a .... = 2-9978 
,, ,, „ directly . . . . = 2-8776. 
(In the case of the “foreheads” of Crabs, the mean error from cr was 3-8028, and 
directly 4-4087. This divergence between the mean error, as found practically from 
second and first moments, is a very good test of the asymmetry of the frequency- 
curve. In the very symmetrical measurements of “ Crabs No. 4,” the modulus, as 
calculated from the standard-deviation and from the mean error, had the near values 
3-874 and 3-863.) 
The curve obtained from the observations as a single group {i.e., d = 16-1914 and 
cr = 3 - 7572 ) is given in fig. 4 (Plate 4). 
Taking y = iVPa have for the fundamental nonic and its first difierential 
/(x) = X® 
+ 24-177,940,535x^ 
+ 1-675,748,344x^ 
+ 299-620,303,770x^ 
— 943-393,909,962x'*^ 
— 864-540,147,350x® 
— 274-750,163,918x^ 
— 34-486,278,563x 
~ 1-394,286,418 = 0. 
* These results were calculated to a higher degre 
rendered necessary by the apparent sensitiveness of 
of the coefficients of the nonic. 
t Mean error is here used, not in Gauss’s sen 
= '7979 a theoretically. 
/' (x) = 9/ 
-f 169-245,583,743x® 
+ 10-054,490,066x^ 
+ 1498-101,518,851x^ 
- 3773-575,639,850x2 
- 2593-620,442,052x~ 
549-500,327,835x 
- 34-486,278,563 
of accuracy than in the case of the Crabs, a result 
he roots in this case to a slight change in the value 
:, but in the sense of arithmetically mean error. 
