MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 305 

 We can now, remembering that 



write down the standard deviations and error correlations of the algebraical constants 



2 /i = 17078, R, re = - -6835, 



S= 4-4773, R,,, = - 7088, 



S, = 18-1898, II,,,, = '9798, 



?. = 4-5840, R,,, = -9980, 



R,= '8129, 

 ft,, = -8340. 



Here h marks the position of the origin of the curve, and the numerical values are 

 only retained to four places of figures, although, of course, in the further calculations 

 the logarithms of the full values of the 2's and R's have been used. 



It will be noticed at once that though a, r, and v have very considerable probable 

 errors, the correlation between them is very high. In other words, as the curve 

 approaches its limiting shape, a, r, and v may vary very considerably, but owing to 

 their close correlation this will not sensibly affect the geometrical shape of the 

 curve. 



The next stage was to determine the standard deviations and error correlations 

 of certain subsidiary constants. Here, as in the determination later of the like 

 quantities for the " physical " characters, we found the umbral notation of great 

 service. It consists, as we have seen, in writing down a .difference equation between 

 any constants, and then replacing the differences Bu by 2,,x,,> $v by 2^, &c., whore 

 X"-> X> & c> > are quantities which obey the relations ^; = 1, ^; = 1, X ,,X B = R,,,. Thus, 

 if tan< = v/r, we find for the umbral equation 



<? v - -" v - 

 -^A* ~~ , A." 



Whence, putting in the numerical values, we have 



2^X4 = Autl. 1-163,2J 15 Xll - Antl. 2-933,070G x , 



where Antl. stands for antilogarithm, in which form we found it easiest to keep the 

 umbral coefficients. The square of this result gave at once 



$t= -087,926 



and dividing out by its logarithm, we have the pure umbral equation 

 x ^ = Antl. -219,0937x, - Antl. r988,9728 x ,.. 



VOL. CXCI. A '2 R 



