xgoS.] JENNINGS— HEREDITY IN PROTOZOA. 399 



The iiican index given below the tables is the mean of the quotient 



• : it shows essentially what percentage the breadth is of the length. 



length 



This mean was found, without computing the index for each individual, by 



the following formula : 



2 = ^(1 + Cl'->CbCl). 



-^ L 



Where i is the mean index, As is the mean breadth, At the mean length, 

 Cb the coefificient of variation for breadth, Cl the same for length, and r is 

 the coefficient of correlation between length and breadth. 



I am greatly indebted to Dr. Raymond Pearl for assistance in the mathe- 

 matical treatment of the data. 



The resnlt.s of the meastirements of a random sample of 400 of 

 Ctilture I are given in Table I. 



It is evident on inspection of this table that the individuals fall 

 into two well-marked groups, one set varying in length from 84 to 

 144 microns, the other set varying from 164 to 240 microns, while 

 between these groups, in the region from 144 to 164 microns, only 

 two specimens are found. The mean length for the entire sample 

 falls at 165.840 microns, almost precisely in the region where no 

 specimens are found. The smaller set have their mean length at 

 125.420 microns: the larger set at 200.972 microns. 



These results are shown as frequency polygons in the lower por- 

 tions of Diagrams i and 2. 



4. Method of Constructing the Polygons. 



In making the polygons for length, three units of measurement (12 

 microns) were grouped together to make a single unit of the abscissa of the 

 polygon. This was done in order to destroy any irregularities due to un- 

 conscious prejudice on the part of the measurer for certain numbers. Thus, 

 in measuring a large number of individuals, it may be found, for example, 

 that few are recorded at 51, while at 50 there are many; or the reverse may 

 occur. This is due only to the fact that in doubtful cases falling between 

 these numbers the measurer unconsciously gives the preference regularly to 

 one of them. The error thus introduced is extremely small (it can hardly 

 be more than one micron in any case), but if the polygon is made without 

 grouping together adjacent classes, there appear extreme irregularities in 

 its outline, irregularities that are quite without significance. When three 

 units are thrown together, any marked irregularities remaining in the poly- 

 gons are almost certainly due to peculiarities in the material itself. It is 

 of course possible that small peculiarities really existing may be hidden in 



