-'34 PROFESSOR K. PEARSON AND MR. L. N. Q. PILON 



A/i w , Ac,, Ac.,, . . . Ac,,, have been made in the determination of the frequency con- 

 stants. In other words, we have here the frequency distribution for errors in the 

 values of the frequency constants. 



(3) Conclusions to be drawn from the, form qf(vi.)._ 



(a) The distribution of the errors of frequency constants, if treated exactly, will 

 generally be skew, for the cubic and higher terras in the A's do not vanish. If, 

 however, the cubic terms are small as compared with the square terms, the frequency 

 distribution of errors will approximate closely to a normal correlation surface. 



It would be impossible to evaluate the remainder after the second power terms in 

 TAYLOR'S series for any general expression f (#,, x 2 , . . . X M ; c,, c 2 , . . . c y ,) for frequency. 

 In special cases we have found that terms of the third order amount in the most 

 unfavourable circumstances to 4 per cent, of the terms of the second order, generally to 

 a good deal less.* Probably the series in most cases converges with considerable 

 rapidity. The fact, however, that we are dealing with the first terms of a series should 

 be borne in mind. It does not seem to have been sufficiently emphasised when the 

 probable error of the standard deviation is taken to be G7'449/ x /2>Tper cent, of the 

 standard deviation. The usual proof of this result, however, involves the same 

 assumption as to the smallness of the cubic terms. 



(/3) Supposing the errors so small that we may neglect the cubic terms, we 

 conclude that the errors made in calculating the constants of any frequency distribu- 

 tion are 



(i.) Themselves distributed according to the normal law of errors, 

 (ii.) Correlated among themselves. 



Both these conclusions are of the utmost importance. The first enables us to 

 obtain the probable errors of the frequency constants ; the second depends upon the 

 fact that C,/, G,,, and F M , are in general not zero. The standard deviations of, and 

 the correlations between, the frequency constant errors can now be calculated by the 

 ordinary theory of normal correlation. 



Before, however, proceeding to these calculations, we may draw one or two other 

 conclusions of considerable generality and wide significance. 



(y) Consider a race fully defined by the variations o-,, o-,, o- 3 , . . ., &c. of the organs 

 of its members and their correlations r 12 , r. a , r n , .... Now let a random small 

 selection be made of this race, defined by 



o-! -f- So-,, o-, -f- So-,, o-;, + 80-3, . . . r r , + Sr,,, r, 3 -f- Sr, 3 , r, 3 + Sr, 3 , . . ., 

 where the magnitudes of "So-!, So-,, So- 3 , . . . 8r K , Br- a , Sr 13 , . . ., are quantities depending 



* See, for the relative order of two terms of the second and third orders, " Regression, Heredity, 

 and Panmixia," ' Phil. Trans.,' A, vol. 187, p. 266. 



