70 Coojyerative Investigations on Plauts 



sub-environnients inay have altered in an aibitrary nianner the means of the 

 ditferent parent plants. For exainple, with oiily a lew parents somo inay have 

 been more highly t'avoured by light, soil or water than olhei-s. At any rate this 

 method is valiiable for purposes of control, although as it involves the labour of 

 finding individiial offspriiig ineans, it cau oiily be occasionally applied. 



Thirdly we rnny proceed by the homotypic relationship. This reijuires a brief 

 theoretieal treatment. 



Let^ be the mean charactcr iu any iiulividiiai parent, P be the mean parent; 

 the mean character in any offspring, the mean offspring and R parental 

 correlation ; then if o-^ o-j, be the standard-deviations of parent and offspring 

 respectively, and iV the total number of cases, we have 



^^sip-pno-0) 



But if we weiglit the offspring as on p. Gl) with the capsule nuinbers of each 

 individual, we shall have 



R = 



N'<T„a-o 



where ?! = number of capsules in an offspring plant, N' = S{n), o-p' = Standard 

 deviation of parents and aj of offspring weigiUed witli capsule numbers. 



If now we use c to denote- the capsule, we have ?!o=S(c), where S is the 

 summation for every capsule of an individual. Further 



0=S (no)lS (n) = S% {c)lN = C, 



where C is the mean capsule of the whole series of offspring. Heuce we find 



S{p-P)t{c-C) 



R = 



N'ffJ <7o 



S'(j>-P)(c-C) 



•(V), 



where S' is a summation of every parent jjlant and offspring capsule. But if r be 

 the correlation between parent plant and offspring capsule 



Hence it follows that 



^^S'(p-P)(c-C) 

 iV'o-p'o-c 



R=^r» (vi). 



0"o 



• Multiplying by irjlap' we have the regression coefiSoicnt equol to ivJiTp the result of cur first 

 method. 



