Researches into tlie theory of probability. 



37 



and, as = = x^ we now only want an equation for x, which is 



y I y -2 — 2 (^/i + y^\ = C3 , 



and the problem is solved. 



This method is applical)le, whenever the collective object consists ol^ a mixture 

 of two races (types), the mean value of the character in question being known for 

 each of these types. 



2:o Suppose the given frequenctj curve to he symmetrical. This case has been 

 treated by Pearson (1894). It is found that the two components are then either 

 symmetrically situated to the mean and possess the same number of individuals, 

 or that the two components have the same mean, coinciding with that of the fre- 

 quency curve. In either case the solution is found through elementary operations. 



3:o Suppose the ttvo components to possess equal standard deviations. 



Using the same abbreviations as before and putting 



t^^l — 



we now have the equations 



h^^^ + 63^2 = 0, 



(47) 



from which equations we may eliminate z^, .e-^, ?;j and l).y The resulting equatioa 

 for t is then 



(48) 2/« + C,^ + C^=U. 



When this equation is solved, we find and 60 to be the roots of the 

 quadratic 



49) f^^yJ^t^i). 



Finally the values of and z.^ are found from the two first equations (47). 



The supposition here made — that 0^ = 0^ — is of a more general character 

 than those made in l:o and 2:o. Especially in biology it is a fairly probable sup- 

 position that two types found together in the nature often possess nearly equal 

 standard deviations. We may then use this method to separate the two components. 

 We find for instance that the 19 pure lines of Phaseolus vulgaris cultivated by 

 Johannsen (compare table V) possess standard deviations that are surely not iden- 

 ^tical, but yet are of the same order. As an instance I have applied this method 

 to the same curve, to which Peaeson first has applied his general method, namely 

 the distribution of the frequency in the breadth of the head of 1000 Neapolitan 

 crabs, measured by Weldon. 



The equation (48) gave here, using the values of the nioments obtained by 

 Pearson, 



