PROF. K. PEARSON ON THE MATHEMATICAL THEORY OF EVOLUTION. 95 
it is easy to see that 
3 ^i) (1 - {- ej)] 
is less than 
^ {— fj) {1 - \/{— ^i)}. 
Hence, our final conditions are 
v/{— ^i) > 
then 
but if 
then either 
or it must lie between 
and 
(- > G (- ei) (1 + y (- ej)} ; 
(— ^2) > 6 (— ej) {I ^ 61)}, 
3 (- ei) {1 — (— ^ 1 )} 
G (— ^j) [1 + \/(— ^i)}- 
(iii.) ej positive and Cj negative; if the values of lo are real, one must be negative, 
and therefore the solution impossible. 
(iv.) negative and 63 positive; if the values of w are real, one must be negative, 
and therefore the solution impossible. 
Thus we conclude : 
If the excess and defect are not zero, the frequency-curve, although symmetrical, is 
not normal. If the excess and defect are of opposite signs, then the frequency-curve 
cannot be broken up into the sum or difference of two normal curves with common 
axis. The frequency-curve, if compounded of normal-curves at all, is of a higher and 
more complex character. If the excess and defect are of the same sign, then, 
provided certain relations hold between the numerical values of the excess and defect 
given in (i.) and (ii.) above, there is a real solution of the equation which resolves the 
frequency-curve into two components. 
(13.) I propose to illustrate this discussion by the consideration of a numerical 
example. Professor Weldon has kindly complied with my request for the numerical 
details of the most symmetrical curve deduced from his measurements of Naples 
crabs by placing the following statistics for a shell measurement—No. 4 of his series 
—at my disposal. The resultant-curve and the corresponding normal curve are 
pictured in fig. 3 (Plate 3). Clearly, from the ordinary statistician’s standpoint, we 
could not expect a more symmetrical result, or a closer graphical agreement, with the 
normal curve. But is this a real or merely an apparent agreement? The answer is, 
as we shall see, vital for the interpretation to be put on Professor Weldon’s results. 
