PROF. K. PEARSON ON THE MATHEMATICAL THEORY OP EVOLUTION. 105 
observations, diverges considerably from the observational result, namely (see fig. 4), 
in the defect of carapaces about 45. This defect largely contributes to the 
asymmetrical appearance of the curve. I felt very confident that by neglecting 
the eccentric group of “ giants ” I could find two components, whose resultant would 
fit the curve of observation as closely as the resultant-curves found for the similar 
case of the forehead of Crabs. I was peculiarly interested, however, in ascertaining 
whether the method of resolution by aid of the nonic would pay more attention to 
the outlying giants or to the less improbable defect of individuals about 45. I even 
imagined that out of the nine possible solutions some might be solutions for the 
giants and some for the 45 defect. As a matter of fact, the two solutions which 
have any meaning are entirely taken up with the very improbable outlying eccen¬ 
tricities of the observations. These eccentricities must first be removed from the 
observations before the method will be of service in resolving the asymmetry of the 
bulk of the observation-curve. 
The method in which the nonic deals with the abnormalities is very characteristic, 
and I venture to think highly suggestive. 
In fig. 4 the normal curve excluding the two giants is given. It fits the observa¬ 
tion-curve, as far as appearcmces go, slightly better than the true normal curve. 
But the first solution of the nonic tells us not to absolutely reject the giants. It 
gives us two components, the first of which fits the observations slightly better than 
the normal curve D (giants excluded). It has practically the same area (995‘86 as 
compared with 996), a slightly less standard-deviation (3'5595 as compared with 
3*6051), and consequently an increased maximum ordinate. This, with a slightly 
shifted axis, gives a somewhat better fit. In addition to this first component we have 
a second component with an area of 2*140, and a mean of 70 for the carapace. This 
component corresponds closely to the tivo giants with a mean of 67. It has, how¬ 
ever, an imaginary standard-deviation. Clearly the addition of two to the first 
component, if distributed really, could make no sensible change in its appearance, 
and we may then sum up the first solution of the nonic in the followdng words :— 
It does not absolutely reject the two giants, but places an imaginary distribution of 
2*14 in their neighbourhood, and thus obtains for the other component and the 
resultant-curve (which must be practically identical with it) a better approach to the 
observation-curve than if the giants had been rejected. 
It would appear, therefore, that our method of dissection offers, by means of 
small components with imaginary distributions, a means of obtaining better results 
than by simply injecting (or, perhaps, even weighting) anomalous observations. 
The second method by which the nonic attempts to account for the eccentricities of 
these carapace measurements, is by mixing a small joopulation of about 2*7 per cent, of 
giants with the normal population. These giants have a mean carapace of 48*5, while 
the rest of the population has a mean of only 43. This population of giants, however, 
has a very large standard-deviation, i.e., 8*9330 as compared with the 3*38!)7 of the 
MDCCCXCIV.—p 
