PROF. K. PEARSOR ON THE MATHEMATICAL THEORY OF EVOLUTION. 89 
parative smallness of the number of individuals dealt with, and the resulting fact 
that the observation-curve can at best only be an approximation to the true 
resultant. 
2nd Solution .—Precisely similar calculations were undertaken for the value 
y), = — 6’724, and it will, accordingly, be sufficient to cite the final conclusions 
here. 
Quadratic for y Y — ■3412y — 6‘724 = 0. 
1st Component. 
2nd Component. 
Cl = 467-2, 
C3 — 
532-8. 
= 2-769, 
— 2-428. 
0-1 = 2-878, 
o’a = 
4-7702. 
yi= 64-764, 
2/3 = 
44-559. 
These component-curves are drawn in fig. 2, and their ordinates added together. 
We see that we have again broken up our asymmetrical frequency-curve into two 
probability-curves, whose sum is a very close approximation to the original curve. 
2>rd Solution : p .2 = 4•170. 
While the first two solutions have been additive, this solution makes y^ and y^ 
(^2 = 7172) of same sign, or the centroids of the component curves fall both on the 
same side of the centroid vertical of the frequency-curve. Accordingly the area of 
one of them must be negative, and the solution promised to be a subtractive one, i.e., 
to represent the frequency-curve as the difference of two normal curves. 
Determining P3 and then from (27), we find = — 3'605 ; hence 
/ -f 3-605y + 4-170 = 0. 
The roots of this equation are, however, imaginary. In the case of crabs’ foreheads, 
therefore, we cannot represent the frequency-curve for their forehead lengths as the 
difference of two normal curves. 
(10.) So far as the nonic is concerned, our work is now accomplished. Taking the 
biologist’s measurements and assuming them to be the chance distribution of two 
unequal groups about two different means, then one or other of our solutions is the 
correct answer. Applying the test of the sixth moment, we find for the observations 
fig = 177,004, while for the first solution it is 188,099 and for the second solution 
192,446, According to this test, the first solution is the required one,'" but, as we 
have noticed, the two solutions are themselves much closer together than either to 
* The theory of correlation will here, perhaps, confirm this result. Professor Weldon tells me that 
the first and not the second solution is in good accordance with his other measurements. 
MDCCCXCIV.—A. N 
