loo PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
These stages seem to rej)resent the mathematical treatment of this portion of the 
problem of evolution. 
(15.) Although the nonic corresponding to “Crabs No. 4,” has no real negative 
root, I found on tracing its value for values of between 0 and — 2, that near 
P 2 = — '82 it reached a minimum value of about 199 as compared with about 1761 
at 0 < 1254 at — 2. Here then was, as it were, a tendency towards a root, and 
the question occurred to me whether this “ tendency ” in any way corresponded to 
the groups into which the “foreheads" were differentiated. I therefore investigated 
the root of the first derived function of the nonic lying about — ’82, and found it 
to be — '8497. This led to ^9^ from equation (27) being — 5’2521, whence 
or 
Whence nearly 
+ 5-2521y — '8497 = 0, 
= -15705, y3=- 5-40915. 
= -972, = -028, 
or the numbers in the two groujDS are 
Cl ■— 971 and = 28. 
Clearly even this “tendency to a root” in no way fits either solution of the 
“forehead” case, and No. 4 measurements neither break up, nor have they even 
a tendency to break up, in the same manner as the “ foreheads.” Since the nonic 
must always have a “tendency” to two real roots at a time, we may note that the 
other root to which it may be said to tend, or for which /’(ju) is a minimum, lies 
between — '9 and — 1, and is just as insignificant as that investigated above. We 
may say that not only is the material of No. 4 homogeneous, but it has not even a 
“ tendency ” towards heterogeneity. 
III. 
(16.) The object of the present paper being solely to illustrate a general method 
for the reduction frequency-curves to normal types, and not a biological investigation, 
it might suffice to stop at this point, when the rules for the reduction of symmetrical 
and asymmetrical curves have been given and illustrated. But it must be remembered 
that the method depends u])on the solution of a nonic, and that the variety presented 
de])ai'tment of tlie Doubs. IMr. Batesojj lias found a double-bumped curve for the claspers of Earvigs. 
Without the investigation of measurements of another organ, it seems impossible to say whether the 
inhabitants of the Doubs, as Bertillon supposes, are a mi.vture of races, or Mr. Bateson’s earwigs were 
I'eallj homogeneous. In either case our methods of investigation Avould show' the proportions belonging 
to each group of the mixture, or to each gi’oup of the differentiating species. 
