1903.] 



to the Th eory of Evolution. 



303 



Or, noting that 



I = S )n ^> n (ci)l(mn) =S m (c p )/m, we have 



* 



2 



where tr is the standard deviation of the character-individualities free 

 from the position factor. We see that it is precisely the same quantity 

 as we have previously used for the mean standard deviation of the 

 arrays for given positions. 



Next taking the correlation of characters c\ and c' 2 in positions £>i 

 and p-2 we have 



&tn fa' 2 ) + S m (cico) = S m (cr) + S m (cic&) - 2$ m (c x ) ml + m 2 c 2 . 



To get this result we have multiplied every quantity like c\ = Ci - 

 by every other quantity like c 2 = Co - c P2 and by itself, and then added 

 such quantities together for every position on the one organism. Thus 

 on the left hand side there are m terms in the first, m(m - 1) terms in 

 the second summation ; on the right hand side there are m terms in 

 the first, m(m - 1) terms in the second and m terms in the third sum- 

 mation. Now sum for each of the n organisms, and we have 



miner'' 2 + nm(m-l) Eo-' 2 = mn (o- 2 + c 2 ) + nm (m - 1 ) (po- 2 + c 2 ) 



Now while this proof is independent of the theory of partial cor- 

 relation coefficients, involving only simple algebra, and is further 

 independent of any consideration of linear regression, it yet wants 

 something of the width of the former theory, which allows us at once, 

 for example, to correct for a combination of factors, such for example 

 as for both growth and position influences simultaneously. The 

 difficulty lies entirely in the extent within which it is legitimate to 

 assume the relation between position or age, and the mean value of the 

 character at that position or age to be linear. It is therefore clearly 

 advisable to start by plotting this relationship,* and fitting, if possible, 

 such position or growth graphs with appropriate curves. If, for the 

 series of positions dealt with or the period of growth taken, we find that 

 a straight linef is a close approximation to the relationship, then we 



* In the case of some animals and many plants the relationship is in itself 

 of much interest, for it expresses a law of development or growth in serial parts, 

 f The analytical consideration of this point is very simple. If the regression 



- '2m 2 nc 2 + m 2 nc 2 . 



or, as before 



Whence 



(X). 



