BIPARENTAL INHERITANCE IN PARAMECIUM 405 



difference between this value and 1 gives 0.00004758 as the proba- 

 bility that we should have any deviation as great as 9. Dividing 

 the former value by the latter, we find that the odds against so 

 great a deviation as that actually observed are 21,016 to 1, unless 

 there is something in the pairing that causes the two members of 

 the pairs to be more alike in their fate (not less alike, as the theory 

 of sexual differentiation held). 



It would be of interest and value if it were possible to find for 

 the present case some standard of value corresponding to the prob- 

 able error or standard deviation of the common 'normal' curve 

 of probabilities. It may be of interest to examine a curve for 

 such cases as those we are here dealing with. To get such a 

 curve, we must determine the probabilities for each of the possible 

 numbers cf pairs (in the way just set forth); then these may be 

 plotted on some convenient scale. In doing this, one or two sim- 

 ple considerations will aid. If the number drawn is less than 

 half of the total number (that is, if n is less than ^ m), then evi- 

 dently it is possible that all the numbers drawn should be differ- 

 ent, so that we must begin with pairs. Further, the number of 



n n — 1 

 pairs might be — (or — - — , if n is odd), but cannot be greater 



than this. So in such a case we must find the probability for 



fi 

 each number of pairs from to -— . 



I have plotted such a curve for one of the cases described by 

 Miss Cull. She found that after twenty days, 51 lines had died 

 out, from the entire 186; and among these were 13 pairs. In 

 this case the possible numbers of pairs range, in accordance with 

 the considerations just adduced, from to 25. 



Determining by formulas (3) and (4) the probability for all 

 these numbers, we obtain the results given in table 36, page 412. 

 We employ these probabilities as ordinates. while the numbers of 

 pairs are laid off on the abscissa. This giv^es a curve or polygon 

 such as is shown in figure 1 . The sum of all the ordinates is here 

 equal to 1 (or if we prefer we may take the total area of the poly- 

 gon as 1). It is evident that this polygon bears some resemblance 

 to that obtained from distributions following the normal law, but 



