1903.] 



to the Theory of Evolution. 



301 



Let p = mean position, <j p == standard deviation of positions on the 

 arbitrary position scale. Let there be n organisms, and suppose that 

 S, n denotes a summation of all m homologous parts on an organism, 

 and S a summation for all n organisms. Then, if <r p = cr lh = o-^, 



am (in - l)r Pllh <r p 2 = $ n {S ? „ (^i -p) (p 2 -ft)} 



= S % {S m (^i - p) x 8 m (p. 2 -p) - S m (vi -p) 2 } 



But Sm(pi-p) = 0, hence, since $ m (pi-ft) 2 = ma- p 2 , 



nm (m - 1 ) r PiP2 <r p 2 = - nmo-/, 



1 , ... 



w r v*h = -^n ( Vll) - 



Further 



nm (m - 1 ) r PlC2 o- p o- c = S» { S m (pi - p) (c 2 -c)} 



= S w {S m (c 2 - c) x S m -|>) - S m (d - c) (pi -p) } 



= - S w { S m (d - £) (pi - j>) } . 

 But S^jS^^i-^^-jo)} = nmo- c (r p r ClPl . 



Hence 



r A* = - ~r[ = x •jpip. ( viii )> 



a relation precisely similar to that discovered in the case of growth 

 periods for brother's head-lengths from the actual numbers on p. 294. 

 Substituting we find 



Then substituting in (vi) and using (vii) we determine the simple 

 formula for homotyposis corrected for positional differentation 



where r pe stands for the correlation of character and position on the 

 organism. 



An exactly similar formula might be found for the correction for the 

 age or growth factor, if the m homologous parts dealt with had the 

 same distribution of ages or growths in each organism. 



(7.) Now the equation just found has the serious disadvantage that 

 it is based on the linearity* of the regression relation between position 



* The reader should note that this condition does not involve any assumption 

 of normal frequency, or the Gaussian law. The latter applies only to a very 

 special case of linear regression. 



