MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 249 



We are now in a position to write down from (vii.) of Art. (4) the complete 

 frequency surface for errors in the constants of a normal frequency surface for 

 m organs. It will suffice to write a type of each term. We have 



n fl! + RufAo-,)* 2R,r,, Ar, A<r 4 

 P A = P X exponent - c \ --=- -f- H -- - 



~\ ffi<r t 



(A , B)S + . . . + ? AS 4 ,, s + . . . 



+ 2 =***,. + . . . + 2 s-. a,.*., + . . .(xxxi.). 



Now this result again seems at once to give conclusions of considerable importance. 

 Thus : 



(a) Since there are no terms of the type ACT, A/V,, we infer (i.) that the random 

 selection of variation in one organ will most likely only vary the correlation between 

 two other organs by terms of the second order ; and that (ii.) the random selection of 

 correlation between two organs will in all probability only change the variability of 

 a third organ by terms of the second order. 



(/3) The selection of correlation- between any two organs will most probable vary 

 the correlation between a second pair, i.e., terms exist in Ar,., A ;;t , &c. 



(y) The selection of the variation for any organ varies the correlation between 

 that organ and a third organ, and vice versd the selection of correlation between two 

 organs changes the variability of both organs. And lastly 



(8) The selection of correlation between two organs varies the correlation between 

 either organ and any other organs. 



We may exhibit these results more clearly by taking four special organs, say, 

 femur, tibia, humerus, and radius. Then a group having the variability of its femur 

 different from that of the general population, will also have, in all probability, the 

 variability in its tibia, humerus, and radius different ; the correlations femur-tibia, 

 femur-humerus, and femur-radius different; but those of tibia-humerus, humerus- 

 radius, and radius-tibia only slightly different. Further, a group having the 

 correlation of its femur-tibia different from that of the general population, will also 

 have all the other correlations, humerus-radius, femur-humerus, femur-radius, tibia- 

 humerus, tibia-radius, different from the values for the general population. Further, 

 the variability in femur and tibia will be changed ; but in all likelihood the variability 

 in humerus and radius only slightly changed. 



These general conclusions, which seem to cast considerable light on the manner 

 in which selection influences the variability and correlation of organs, must now be 

 reduced to quantitative expression. 



VOL. CXCI. A. 2 K 





