MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 275 



On the other hand, the percentage probable errors of p and y are from (xlix.) 



and (1.) 



1 67-449 // 1\ L . . 



and -J&) v ( l + is) res P ectivel y- 



67449 1 



Here S is equal to the series Bi/p B 3 /p 3 + B /p s ... which tends to zero as 

 p increases. 



The errors in p and y thus tend to increase indefinitely as p increases. It may then 

 be asked how the form of the curve can be determined with any degree of accuracy. 

 The answer is simple : Equation (li.) shows us that the correlation between errors in 

 p and y tends, asp increases, to become " perfect," i.e., unity. But as p increases 

 indefinitely, it has been shown that the frequency curve of this type passes over into 

 the normal form.* It is the high correlation between errors in p and y which 

 renders the curve, when plotted to observations, such an excellent fit. If the errors 

 injp and y were independent, this would not be so. At the same time it renders 

 j) and y unsuitable for tabulation as physical or biological constants of the frequency. 



Turning to (Iv.) and (Ivii.) we see that the standard deviation, cr, and the skew- 

 ness, Sk., are suitable constants for tabulation. Their probable errors do not tend to 

 increase indefinitely with p, and will always be small, if n be large. 



Hence a frequency distribution of this type is best defined by its mean h, its 

 standard-deviation cr, and its skewness Sk. These are constants characteristic of 

 the group, for they are given with small probable errors. If it be desired to draw 

 the form of the frequency-curve, then its algebraic constants, p and y, may be found 

 from 



and the possibly considerable errors in p and y will not vary largely its actual shape. 



(/3.) The nature of the probable errors of the other allied constants may now be 

 considered. The mean and modal frequencies per unit variation of organ, or y^ and 

 y are seen by (Ixii.) and (Ixiv.) to have small percentage probable errors, and are, 

 therefore, good for use as characteristic physical or biological constants. But it 

 should be noted that the modal frequency is considerably more exact for moderate 

 values of p than the mean frequency. For example it would be somewhat better to 

 tabulate the modal than the mean frequency of the barometer as a physical charac- 

 teristic of climate. 



The probable errors of the distances from the mean to the mode arid from the 

 mean to the terminal of the range are given by (Ixxii.) and (Ixxiii.). Since 

 d = \ly = cr/ */ (1 -f p), we may write the first 



oV(d 



+ p 2 (1 + p)S 



* ' Phil. Trans.,' A, vol. 186, p. 374. 

 2x2 



