BIPARENTAL INHERITANCE IN PARAMECIUM 409 



putations become fairly simple. Such tables have been pub- 

 lished by De ^Morgan ('45) (six-place logarithms), and by Degen 

 ('23) (tweh'e place logarithms of )il for all numbers from 1 to 

 1200), but I have not been able to obtain these. Pearl and Mc- 

 Pheters Cll) have recently published a useful table of such sums 

 of logarithms for the numbers from 1 to 100. This table will be 

 found convenient when the numbers dealt with fall within it. 

 For use in the present investigation I have prepared such a table 

 up to 243!. 



But the value of n ! for any number may be found with sufficient 

 accuracy from Stirhng's formula. This, in form for practical 

 use, is as follows : 



n\ = — l/27r 71 



Here: 



e = 2.7182818 (log. 0.4342944819) 



jr = 3.1415927 (log. 0.4971498726) 



V27r = 2.506628 (log. 0.3990899342) 



All the operations should be performed by the aid of logarithms 

 (even where a computing machine is available), the best course of 

 procedure being as follows: 



(1) Find the logarithm of n; multiply this by n. 



(2) Multiply log. 0.4342944819 by n, and subtract this pro- 

 duct from the result of (1). 



(3) To the result of (2) add log. 0.3990899342. 



(4) To the result of (3) add | the logarithm of n. 



This gives the logarithm of the given number n\, which may be 

 used in formula (3). For example 



186186 



^^^- = — T^'^^^TT y 186 = log. 342.8844688 



In using formula (3) it is of course necessary merely to add 

 together the logarithms of the factors in the numerator, and sub- 

 tract from the result those of the denominator; the result is the 

 logarithm of the probabiUty. 



