246 PROFESSOR K. PEARSON AND MR. L. N. G. F1LON 



Attention should be drawn to the fact that we have replaced errors by differentials. 

 This is only legitimate so long as product terms in the errors are negligible as com- 

 pared with linear terms. This is the assumption almost universally made by writers 

 on the theory of errors.* It will not lead us astray, so long as we take care in any 

 practical applications to verify the smallness of Ar,,, ACT,, ACT,, as compared with 

 r, 2 , tr,, and <r., respectively. 



(8) To find the Probable Errors and Error Correlations of the Constants of a 

 Normal Frequency Distribution for Three Organs or more. 



It will scarcely have escaped the attentive reader that our investigation hitherto, 

 only involving two organs, has left several important problems untouched. For 

 example, it has dealt only with the direct effect of random selection. But we may 

 ask such a question as this : Wliat is the change in the correlation of two organs 

 when the variability of a third is randomly selected ? Or again : What is the 

 change in the correlation of two organs when the correlation between one of these 

 and a third, or between a third and a fourth, is randomly selected ? All these are 

 important problems in the theory of evolution. 



The general equation to a normal frequency-surface for m organs is : 



where H is the determinant 



1 ?Y> 'Y; 



r ai 1 r n . . . 



r,, r,, 1 ... 



and R sj , is the minor of the term in the s ih row and s' th column. 

 We require first to find the quantities like (vii.) of our Art. (4). 



log * = log (nj(^r -) k log II - S, (log .) - $8, - S,,, 



._. s 



Bo? (ll w)\ 



* GAUSS, admittedly, ' Theoria CombinatioDis Observatiounm ' .... p. 53, Problema; LAPLAOK and 

 POISSON, actually but obscurely ; see ' Theorie analytique des Probabilites, Liv. II., chap. IV., and 

 ' Recherches sur la Probabilite des Jugements,' chap. IV. ; more cleai-ly in TODHUNTER'S account, 

 'History of Theory of Probability,' Art. 1,002 et seq. Further, CROFTOS, Article 'Probability,' 48, 

 for a like assumption. 



