MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 231 



II. ON THE PROBABLE ERRORS OF A SYSTEM OF FREQUENCY CONSTANTS. 



GENERAL THEOREM. 



(2) Let there be a group of n individuals, for each of whom a complex of m organs 

 is measured, and let z 8x t 8x., . . . 8x m be the frequency with which individuals having 

 a complex of organs lying between x lt x^ + 8x t ; x 2 , x z + Sx 2 ; ... x m , x, a + 8x m> occur 

 in the total group of n. Here x shall measure the deviation of any organ from the 

 mean of all like organs in the group. Let h l} h.,, h. A , . . . h m , be the mean measurements 

 on the organs, so that hi + x,, h 2 + x 2 , . . . h m + #/, is the system giving the actual 

 measurements on any individual. Then h lt h*, h A , . . . h m , are the first set of constants 

 of the frequency ; they determine the " origin " of the frequency surface. 



Let this frequency surface be given by 



Z -~ J \*^l> 3<2i X 3t X m; C i> Clt ^3> Cp)t 



where c lt c 2 , c 3 , . . . c p , are p frequency constants, which define the form as distin- 

 guished from the position of the frequency surface, and which will be functions of 

 standard deviations, moments, skewnesses. coefficients of correlation, &c., &c., of indi- 

 vidual organs, and of pairs of organs in the complex. 



The problem before us is to find the probable errors of the A's and the c's, which 

 constants fully determine the position and shape of the frequency surface. Let 

 o"t, "/v ^i,,, 0X> 0V, > <r v "r-,., be the standard deviations of the quantities 

 7ij, h 2 , . . . h m , and c lt c 2 , . . . c v . Then a knowledge of these standard deviations will 

 give us at once the probable errors of the frequency constants, for we have only to 

 multiply the former by the numerical factor '6745 to obtain the latter. 



Let us now suppose that the value of the frequency constants had been A x + AA,, 

 A 2 + AA 2 , . . . h m -\- A/4, GI + Acj, c 2 + Ac 2 , . . . c p + Ac p , instead of the observed 

 values. 



Then the frequency of any observed individual would have varied as 



f(x l + AA,, x 2 + AA 2) . . . x m + AA M ; c, + Ac,, c 2 + Ac 2 , . . . c p -\- Ac p ) 



instead of as 



/(x,, a; 2 , x 3 , . . . x m ; c,, c 2 , c 3 , . . . c p ). 



Hence on this hypothesis the probability, P A , of the set of individuals observed in 

 the group actually occurring is to the probability, P , of the set occurring when the 

 constants are A lt h 2 , . . . h m ; c lt c 2 , . . . c pt in the ratio of the product of all quantities 

 like / (xj + A/ij, x 2 + AA 2 , . . . x m + AA m ; c, + Ac l5 c 2 + Ac 2 , . . . c p + Ac,) . for all 

 values of x lt x 2 , . . . x m , to the like product of all quantities like f(x\, x 2 , . . . x m ; 

 Cj, c 2> . . . c p ), or 



+ A/I,, x t 



n/(x l ,x t ,.. .-x m ; Cj.cj, . .. c,) 



