90 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
the observations (see p, 75). In fact, the contours of the compound-curve for both 
solutions are very close together, and neither differs more from the observations than 
most normal curves differ from symmetrical frequency-curves in statistical measure¬ 
ments of this kind. 
The contours are so close that, notwithstanding w^e have demonstrated a theoretical 
uniqueness for the solution of the problem (see p. 72, et seq.), we see that, from the 
standpoint of practical statistics, it is possible for the given material to be broken up 
into more than one pair of normal curves. Thus the problem indeed becomes some¬ 
what arbitrary—at any rate till the asymmetry of the frequency-curve becomes much 
more marked than is the case with that of the foreheads of Naples crabs. Indeed, 
although the method adopted leads to only two solutions, it is quite possible that 
pairs of component normal curves might be tentatively found lying in the neighbour¬ 
hood of those determined by the above solutions, which would give resultant-curves 
fairly close to the frequency-curve. Pi’ofessor Weldon had, indeed, found by repeated 
trials one such solution, but this solution differs widely in the third and higher 
moments from the observations; it cannot, therefore, be considered to have the same 
justification as those given by the present theory. Granted that the original obser¬ 
vations represent a mixture of two species varying about their mean according to 
exact normal curves, our method gives tivo solutions, and two only. Without corre¬ 
lated measurements, it might be difficult to discriminate between these solutions—at 
any rate from the standpoint of practical statistics. The perhaps over-fine theoretical 
test of the sixth moment decides for the first solution. 
II .—The Dissection of Symmetrical Frequency-Curves. 
(II.) Another Important case of the dissection of a frequency-curve can arise, when the 
frequency-curve, wfithout being asymmetrical, still consists of the sum or difference of 
two components, i.e., when the means about which the component groups are distributed 
are identical. This case is all the more interesting and important, as it is not unlikely 
to occur in statistical investigations, and the symmetry of the frequency-curve is 
then in itself likely to lead the statistician to believe that he is dealing with an 
example of the normal frequency-curve. It seems to me that wfithout very strong 
grounds for belief in the homogeneity of any statistical material, we ought not to be 
satisfied by its representation by the ordinary normal curve, simply because our 
results are symmetrical and fit the normal curve fairly w^ell. We ought first 
to ascertain whetlier or not they would fit still better the sum or difference of two 
normal curves. This, at any rate, is a first stage to demonstrating the homogeneity 
of our material, although possibly our test for tw'o may fail, not because our material is 
homogeneous, but because its heterogeneity is multiple rather than double.'^'' 
* Symmetry might ari.se in the case of compound frequency-curves, even without identity of the 
means of the components. In this case, for two components we should have for different means, 
