MATHEMATICAL CONTRIBUTIONS TO THE THEORY OF EVOLUTION. 297 



(20.) Probable Errors and Error- Correlations of the Constants of the Generalised 



Probability Curve of Type 



II -)/ ,,->> tan -'(*/) ^pvvi! ^ 



" y n j. //vi \^x.x.u.f. 



i i i v*'/'*/ j 



This curve is discussed at length, 'Phil. Trans.,' A, vol. 186, pp. 376-80. The 

 chief constants are given as follows, if m ('>' + 2), z = v 1 -f- r~, and h denote the 

 origin : 



Moments 



* , , , ,v , ... v 



ju. 2 = ~^~~ t r: ('" + v) (cxxm.), 



4ft :l i/ (/- + !/'-) . . , 



/".i = <i: , /,._]) ( r _ 9) (CXXIV.), 



Ll.j ^^ 7 1~T7 O\ ' ^\ ..... ( CXXV. J. 



Distance of centroid from origin = av/i- .... (cxxvi.), 

 Size of mean organ = /i (cxxvii.), 



Size of modal organ h -} (cxxviii.), 



Distance from mean to mode = d = -- . . . (cxxix.), 

 Skewness = - ^ /y/hjA . . (cxxx.), 



Standard deviation = o- = r.- \ x /(/ 1 - + v') . . (cxxxi.), 



>' vv ' 



y = -'- e- v "/G (r,v) (cxxxii.), 



where 



and 



G(r,v)= 



G (r, v) = r 



is the formula of reduction. 



VOL. CXCI. A. 



. (cxxxiii.), 



2 Q 



