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



(4.) To find the correction to be made to the apparent homotypic correlation, 

 wJien each pair of homotypes is differentiated by a common period of growth 

 from other pairs of homotypes. 



The solution of this problem may be deduced at once from equation 

 (iii) of the preceding problem by simply putting t\ = t 2 . In this case 

 /• = /, r = 1, and we find 



i< <: t (v)- 



1 — r* 



This equation was given by me in a note in Biometrika, vol. 1, 

 p. 404, and its use illustrated on Dr. Simpson's data for Paramec- 

 ium caudatum. 



(5.) To find the correction to be made to the apparent homotypic correlation 

 when the pair of homologous parts are differentiated from each other by situa- 

 tion on tJie organism. 



We have only to put in formula (iii) on p. 290, h = pi and t 2 = p 2) 

 the positional co-ordinates of the first and second homologous parts, to 

 make that formula available for position instead of age differentiation. 

 If we denote by Ci and c 2 the characters of the parts in the positions p\ 

 and p 2 respectively, our solution takes the form below, where we have 

 confined our attention to the same character, 



j> = p^-W 2 ) 



- '2''lh<-'/j>\'--> ~ r PiPa( r Pi''i" + V P\C2 2 ) mmm ^yj^ 



1 _ r p^2 2 ~ r Pl*l 2 ~ ''PlCi 1 + 2 r PlP2 r PlCi r P-2<2 



This follows since r PlCj = r P2Ci , and r lhCi = r Paf:i . 

 We have again, therefore, to find four correlation coefficients. But 

 this formula simplifies immensely if we observe the following con- 

 ditions : 



(a.) Take the same number of homotypes or homologous parts from 

 the same positions in each organism. 



(b.) Enter each one of these homotypes or homologous parts with 

 each other on the same organism, so as to obtain a symmetrical table, 

 i.e., Ci is first entered with c 2 and then c 2 with Ci, 



These conditions are or can be usually satisfied in any homotyposis 

 investigation. 



(6.) Further, the positions will, as a rule, be arranged in series and 

 may be numbered 1, 2, 3, 4, . .m, if m homotypes or homologous parts be 

 taken from each individual organism. The position scale is, of course, 

 perfectly arbitrary, and has nothing to do, for example, with the 

 actual distances between positions on the organism. We can make it a 

 uniform numerical scale, which for convenience we can take to be the 

 same serial order as that of positions on the organism. 



