BIPAREIifTAL INHERITANCE IN PARAMECIUM 461 



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APPEXDIX 

 FORMULAE USED IN THE ANALYSIS 



For convenience of reference there are brought together here 

 the formulae and rules employed in the foregoing paper, for analy- 

 zing the question as to the number of pairs included, when a cer- 

 tain number survive (or die), out of a number of paired individ- 

 uals. In these formulae: 



m = the total number of individuals 



I = the total number of pairs | so that ^ = -rj ) 



n = the number of individuals drawn 



k = the number of pairs obtained 



.r = the probability for any given number k of pairs 



1. To determine the average number (k) of pairs (and for all 

 practical purposes the most probable number of pairs), to be 

 obtained when n individuals are drawn from ?n individuals form- 

 ing I pairs: 



/t = i"i^-i^ (1) 



m — 1 



The integral number nearest this average is (with rare excep- 

 tions) the most probable number of pairs; where this is not the 

 case, it differs by but 1 from the most probable number. 



2. To determine with certainty, for even numbers of n, the most 

 probable number (/c) of pairs: 



(" + 1) (.T+O 



k = A:;^ L (2) 



where the integral portion of the result is the most probable 

 number of pairs. 



