108 PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 
Germans. In other words, the B,ow-Graves contain a mixed population, one element of 
which corresponds closely to the modern South-German po^mdation. Ammon’s state¬ 
ment, therefore, that an evolution has taken place in this particular skull index appears 
to fall to the ground. The whole problem of the compound nature of skull frequency- 
curves, both in England and Germany, is a very interesting and difficult one, and I 
do not wish at present to anticipate results, which I hope when my investigations 
are complete to publish as a whole. The above may suffice to indicate the range of 
problems to which a resolution of asymmetrical frequency-curves into normal 
components may be applied. 
(2.) With regard to the method adopted in the memoir itself, I am very conscious 
of the defects under which it suffers—the laborious character of the arithmetic 
involved, and the question of what may be the probable error of the solution obtained 
by the method of higher moments. But I had to deal with the fact that the problem 
is one which urgently needed a solution in the case of both economic and biological 
statistics. Better solutions than mine may be ultimately found, but although more 
than one mathematically trained statistician has for some time recognized the impor¬ 
tance of the problem, no solution, so far as I am aware, has hitherto been forthcoming. 
With regard to the amount of error introduced by the use of higher moments, a 
word may be said. I have not been able to work out the general problem suggested 
to me by Professor George Darwin : “ Given the probable erro)’ of every ordinate 
of a frequency-curve, what are the probable errors of the elements of the two normal 
curves into which it may be dissected ? ” 
I can, however, indicate the sort of differences which are likely to occur in results 
based on high or on low moments. Suppose the distribution of an organ in a group 
of animals actually does follow a normal frequency-curve. Then it is obvious that in 
selecting 1000 of these animals at random and measuring their organs, an error of the 
same magnitude in the frequency of an organ of a given size is more likely to occur in 
a size near the mean than in a size far from the mean. Now a low moment pays 
greater attention than a high moment to an error in the frequency near the mean 
and less attention than a high moment to one far off. In other words, a frequency- 
curve calculated from low moments fits best near the centre; one calculated from 
high moments fits best near the tails of the observation-curve. The problem is 
accordingly the following : an error in frequency near the tail is not as probable as an 
equal error in frequency near the mean ; but if it does occur a high moment pays 
much more attention to it than a low moment ; on the other hand, the low moment 
pays more attention than the high moment to more probable errors in frequency. 
Which tendency on the whole will prevail ? 
Turning to the result in the foot-note, p. 92, we have for the 2r‘'^ moment— 
Mo, = (2r - 1) (2r - 3) . . . 5.3.1 o-'-'c, 
and 
Mo, = S Bx). 
