PROF. K. PE ARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 99 
This root was now localized. Putting = iw/x" original nonic, I easily 
found X between 0 and I, then between ’15 and ‘16, and by a succession of 
approximations to be '1533, and finally •15326. 
Thus 
p.;^ — P5326. 
p)^ was then ascertained from equation (27) of p. 84, and finally p^ = 'pjp^ was 
found to be 2'17245. The quadratic (28) for y was then : 
y' — 2-l7245y + 1*5326 = 0, 
which has both its roots imaginary. 
Thus, considerably to my surprise, but greatly to my satisfaction, it was demonstrated 
that tliere is ]io solution whatever of the problem of breaking up the curve of No. 4 
measurements into two normal components. 
All nine roots of the fundamental nonic lead to imaginary solutions of the prohlem. 
The best and most accurate representation of No. 4 is the normal curve of fig. ,3. 
The result of this investigation seems to me most important. Professor Weldon’s 
material is homogeneous, and the asymmetry of the “ forehead ” curve points to a real 
difierentiation in that organ, and not to a mixture of two families having been 
dredged up. 
On the other hand, I cannot think that for the problem of evolution the dissection 
of the most symmetrical curve given by the measurements is unnecessary. There 
will always be the problem : Is the material homogeneous and a true evolution going 
on, or is the matei’ial a mixture ? To throw the solution on the judgment of the eye 
in examining the graphical results is, I feel certain, quite futile. 
Whenever in measuring a series of organs the results give an asymmetrical curve, 
we must accordingly proceed as follows :— 
Stage (i). —Break up this asymmetrical curve into components ; if there are several 
solutions, the theory of correlation or tlie test of the sixth moment will, perhaps, 
enable us to say which is the most satisfactory. 
Stage (ii). —Endeavour to break up the most symmetrical curve ; if it cannot be 
broken up, either into normal components with non-coincident axes or normal com¬ 
ponents with coincident axes, the material is homogeneous and the asymmetrical curve 
points to a true differentiation in the organ to which it refers. If, on the other hand, 
the most symmetrical frequency-curve does break up, then if the numbers in its 
component groups be the same (or practically the same) as in those corresponding to 
the asymmetrical curve, we are really dealing with a mixture of heterogeneous material, 
and we shall have ascertained the proportions of the mixture. If the numbers 
should not be the same, then we cannot assert that we have a mixture, but we have 
found a case of differentiation in both oro'ans at the same time.* 
o 
* BEiti’iLLON has found a double-humped fi’equencj-curve for the height of the inhabitants of the 
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