MATHEMATICAL CONTRIBUTIONS TO THE THEORY OP EVOLUTION. 303 



x = a . 2 - in the equation to the curve. Writing tan <f>' = v/(r + 2) and y/ the 

 same function of r + 2 that y is of r, we have 



r + 1 

 Since - . -- is greater than ^/r, and $ and Y are less than < and x respectively, 



it follows that y., is greater than y l} as it should be. Further we find 



/ sin (') - . (clxiv.). 



Here c,' and c 2 ' are the same functions of r + 2 and <' that c, and c 2 are of r and <. 

 The usual process of squaring and introducing the standard deviations into the square 

 terms and the product of standard deviations and correlations into the product terms 

 will give us 2?,,. 



(21.) Illustration. Stature of Children. 



In order to illustrate the difficulties which may arise in determining the probable 

 errors of the constants and the error correlations, we have selected for this illustration 

 not a curve markedly skew, but one which is extremely nearly normal. The problem 

 in this case is accordingly the following one : Are the values of the constants obtained 

 for the distribution and distinguishing it from a normal distribution really significant ? 

 The difficulties which arise in the course of the arithmetical work depend upon the 

 fact that, as the distribution is nearly normal, its constants approach the values at 

 which the type of the skew curve passes over into the normal curve, and conse- 

 quently not only will their probable errors be large, but, as in all cases of approach 

 to limits, they will depend upon expressions tending to become indeterminate. Thus 

 in the evaluation of the determinant A and its minors, we at once found our results 

 depended on the ratio of the differences of very small quantities. We were accord- 

 ingly in this case obliged to calculate our constituents to a degree of accuracy which 

 will, in general, be quite unnecessary, and which was only possible and straight- 

 forward owing to the ready help of a large sized Brunsviga. That the method, even 

 in a critical case of this kind, gives correct results is evidenced by the agreement of 

 our values of the constants with those (probable errors of mean and standard 

 deviation) which can be readily calculated by other processes. 



The example we have selected is that given for the stature of 2192 St. Louis 

 school girls of 8 years of age in ' Phil. Trans.,' A, vol. 186, p. 386. 



The equation to the frequency curve is 



