PROF. K. PEARSOX OX THE MATHEMATICAL THEORY OP EVOLUTION. 75 
pi’actical value. For after we have made the areas and first five moments of two 
curves identical, their sixth moments will in general be (like their contours) much 
closer together than either are to that of the curve of observations. Added to this 
the great labour involved in the calculation of the sixth moment is sufficient to deter 
the practical statistician, if any other convenient mode— e.g., results of measurement 
on other organs—suffices in the particular case to discriminate between the solutions 
found. Thus, while the mathematical solution should be unique, yet from the 
utilitarian standpoint we have to be content with a compound curve wdiich fits the 
observations closely, and more than one such com230und curve may arise. All we can 
do is to adopt a method which minimizes the divergences of the actual statistics from 
a mathematically true compound. The utilitarian jiroblem is to find the most likely 
components of a curve which is not the true curve, and would only be the true curve 
had we an infinite number of absolutely accurate measurements. As there are 
different methods of fitting a normal curve to a series of observations, depending on 
whether we start from the mean or the median, and proceed by “ quartiles,” mean error 
or error of mean square, and as these methods lead in some cases to slightly different 
normal-curves, so various methods for breaking iq3 an abnormal frequency-curve may 
lead to different results. As from the utilitarian standpoint good results for a simple 
normal curve are obtained by finding the mean from the first moment, and the error of 
mean square from the second moment, so it seems likely that the present investigation, 
based on the first five or six moments of the frequency-curve, may also lead to good 
results. While a method of equating chosen ordinates of the given curve and those of 
the components leaves each equation based only on the measurements of organs of one 
size, the method of moments uses all the given data in the case of each equation for 
the unknowns, and errors in measurement will, thus, individually have less influence. 
At the same time it would be of great interest to discover whether other methods of 
dissection lead to results identical or nearly identical with the method of moments 
adopted by the present writer. Any other method analytically jjossible has not yet, 
however, occurred to him ; nor any criterion for distinguishing practically between 
two solutions so close as those of figs. 1 and 2, other than that adopted by Professor 
Weluon when he ajj^^eals to the measurements of a correlated organ. 
(2.) In the case of a frequency-curve whose comiDonents are two normal curves, the 
complete solution depends in the method adoj^ted in finding the roots of a numerical 
equation of the ninth order. It is possible that a simpler solution may be found, but 
the method adojjted has only been chosen after many trials and failures. Clearly 
each component normal curve has three variables : (i.) the position of its axis, (ii.) its 
“ standard-deviation” (Gauss’s “ Mean Error,” Airy's “ Error of Mean Square”), and 
(iii.) its area. Six relations between the given frequency-curve and its component 
curves would therefore suffice to determine the six unknowns. Innumerable relations 
of this kind can be written down, but, unfortunately, the majority of them lead to 
T. 2 
