302 Prof. K. Pearson. Mathematical Contributions [Jan. 20, 



and character. But while organic and homotypic correlations give for 

 a surprising variety of cases sensibly linear regression relations, the 

 relation between position and mean character is far more rarely linear. 

 We obtain, as a rule, remarkably smooth curves. We, therefore, 

 require some modification of equation (ix). 



Still supposing the regression of character and position linear, we 

 should have, if <r' be the mean standard deviation of an array of organs 

 in the same position, 



a-'- = q- 2 (1 - r pc 2 ). 



But if tr M be the standard deviation of the means of the arrays, we 

 have from first principles 



9 9 , '9 



<T- - CT M - + 0-- 



Hence o- M = o- x r pc . 



We can now write equation (ix) in the form 



T? — a °"' 2 + °m 2 M 



This is quite free from r pc , and, what is more, although we have deduced 

 it from (ix) and the relation cr' 2 = cr 2 (1 - r pc -) peculiar to linear 

 regression, it is now free of any limitation as to the nature of the 

 relation between position and mean character. Thus (x) is a far more 

 important formula than (ix), and should always be used, until we have 

 shown that the relation between position and mean character is 

 sensibly linear. If anything, it involves less arithmetic than (ix). 



We can show this ah initio as follows : — Let the individuality of the 

 organism in any homologous part be measured by its excess above 

 (respectively defect below) the mean value of the character for the 

 homologous part in that position.* Then, if c = element of character 

 due to individuality, and c p be the mean character in any position for 

 the n individuals dealt with, 



Ci = ci-c p , Sa(fi) = nc p , and S n (c{) = 0. 



Hence we easily find 



Sh« 2 ) = Sn(cr)-nc p 2 

 S,/?Sh(ci' 2 ) = SjrtSii (cr) - S m (?i^ 2 ), 



* It might be considered better, if the standard deviations of the homologous 

 parts vary very considerably •with position, to measure the individuality by the 

 ratio of this excess to the corresponding standard deviation. Not only, however, 

 does the use of such a ratio immensely increase the arithmetical labour, which is 

 a possibility, -which of course, we could face, but there is also a question as to 

 ■whether the ratio is really a truer measure of individuality. A full discussion of 

 this important point must for the present be deferred. 



