PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 97 
Since Co > 3ej (I + e^), we see that the roots (37) of our p. 94 are both positive, 
and accordingly it is possible to break up the observation-curve into two normal 
curves with coincident axes. 
Calculating the two values of lu we have 
= 3-50971, 
= 1-01148, 
1^2 
/^2 
whence from p. 93 : 
Cl = — a X '0046, 
C 3 = a X 1-0046, 
0-1= \/ (pa X 3-50971), 
0-2 = {Pi X I'Ol 
or 
Cl = — 5 ;^ 
C 3 = 1004, 
0-1 = 5-134, 
0-3 = 2-756. 
For all practical purposes the second group gives the normal curve (c = 999, 
( 7 = 2‘740) of the set of observations; that a half per cent, of Crabs have been 
removed by selection about the same mean is not large enough to be significant in 
measurements of the kind we are here dealing with. So far, then, we may say that 
No. 4 of Professor Weldon’s measurements cannot be treated as the sum or difference 
of two normal curves having their axes coincident with any substantial improvement 
on the normal curve peculiar to the original group. 
(14.) Hitherto we have used “Crab Measurements No. 4 ” to illustrate the dis¬ 
section of symmetrical frequency-curves, but a little consideration shows at once tliat 
this judging of symmetry by the eye is very likely to be fallacious, and No. 4 may, 
after all, break up into two normal curves with non-coincident axes. Should these 
two curves correspond to practically the same groups as in the case of the “Fore¬ 
heads,” then we shall have demonstrated that the asymmetry of that frequency-curve 
is in all probability due to a mixture of two families in the Naples Crabs and not a 
result of differentiation going on in one homogeneous species. The apparent symmetry 
of No. 4 weighs nothing in the balance, as may be readily tested by adding together 
two normal curves with not widely divergent axes or totals. 
What we have been investigating, therefore, in § 13 is really only the special case 
in which the method of our first investigation would fail, owing to the coincidence of 
the axes of the component normal curves—a coincidence which is improbable d priori. 
I, therefore, proceeded to form the nonic for No, 4, a result which requires only the 
values of /xg, and X- already given.t 
The nonic beinof 
Pi + <^iPi + ^hPi + + (/ 7 P/ + «8P2 + «9 = 0, 
* The nearest whole number is here taken for the Crabs in each group, 
t The arithmetic throughout was of course of a most laborious character. 
MDCCCXCIV.—A. O 
