MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 237 

 Let A be the discriminant : 



where a rs = ... Further, let A,, be the minor of the r tlx row and s th column ; then 



2 2 = A,.,/ A and R r , = A^AS/E,) ( v "'-) 



give the required values of S,. and R,. 8 .* 



Further, the standard deviation of AT;, for selected values of all the other AT/S 

 is 1/v/a,,.. 



This value can often be of service. Thus suppose a considerable number of 

 skeletons found, but that only in a comparatively few cases is it possible to pair 

 together the femur and tibia of the same individuals. Then the variations of femur 

 and tibia in the race will be known with great exactness, but the probable error of 

 the correlation between femur and tibia will be given with close approximation by a 



form like '67449 . . In fact, whenever we have obtained from a large number of 



V"rr 



observations the values of the frequency constants of a race with great exactness, 

 then, using these values to obtain an additional variation or correlation constant from 

 a few observations, the probable error will be of the form just indicated, and not of 

 the form '67449 A,,./A. It is needless, perhaps, to remark that the former is far easier 

 to calculate than the latter. 



III. ON THE PROBABLE ERRORS AND THE COEFFICIENTS OF CORRELATION BETWEEN 

 ERRORS MADE IN THE DETERMINATION OF THE CONSTANTS OF A NORMAL 

 FREQUENCY DISTRIBUTION. 



(5) In order to exhibit more clearly the method of investigation, it is desirable 

 that a simple case be first taken, Accordingly we will start with the following 

 problem : 



To find the Probable Errors and Error Correlations of the Constants of a Normal 

 frequency Distribution for Two Organs. 



Let hi, h, be the means, o-,, ar.>, the standard deviations, r, 2 the coefficient of correla- 

 tion, n the total number of pairs of observations, x lt x 2 , any pair of corresponding 

 deviations of the organs. Then the frequency z dx L dx 2 is given by the surface 



1 s 



o * 



z = 



/ , 



* See " Regression, Panmixia, and Heredity," ' Phil. Trans.,' A, vol. 187, p. 301 (e). 



