INVERTEBRATA, CRYPTOGAMIA, MICROSCOPY, ETC. 945 



The apparatus is so delicately constructed that, if carefully mani- 

 pulated, the errors in measuring will never exceed about 1 per cent. 



The mathematical theory by which the probable extent of the error 

 of counting is determined may be briefly summarized as follows. 

 If n denote the average number of blood -C()ri:)uscles to a given space, 

 the relative frequency of other numbers is expressed by the respective 

 terms of the expansion of e" (where e denotes the base of natural 

 logarithms) divided by the complete value of e". Thus, the pro- 

 bability that k corpuscles will appear instead of n is, 



S\j, = e- 



1.2.3 ... /fe 



Making k = n + A, where A expresses the deviation, positive or 

 negative, from the average number, the above expression becomes 

 a2)i)roximately 



\\ ^ =-—-—— e -in- 

 si -K fj In 



Consequently, the " probable error," that is, the error which in re- 

 peated observations would be as often exceeded as not reached, may 

 be expressed as a function of n, or, calling this error w, 



w = 0-4769 VT«. 



If the ratio of this probable deviation to the average number be 



denoted by w we get 



0-674 

 ft) = . 



In a large number of observations, if w denote the probable error, 

 then, according to the laws of probability. 



An error less than \ u> occurs once in every 7 cases. 



i„ „ „ 4 „ 



1 „ „ „ 2 „ 



„ greater than 1 „ „ „ 2 „ 



„ 2 „ „ „ 5-6 „ 



3„ „ „ 23 „ 



4 „ „ „ 160 „ 



5„ „ „ 1,385 „ 



6 „ „ „ 20,000 „ 



and hence may be directly deduced what reliance is to be placed on 

 the result of a single observation, that is, what approximation to the 

 correct average value may be safely expected when the value of w 

 has been computed from the above formula, suited to the conditions 

 of the particular observation. 



The above formula for w shows thut the probable error expressed 

 as a percentage of the average number decreases in the same pro- 

 portion as the square root of the average number increases. Thus, 

 the value of w is reduced to about 5 per cent, if n is made to equal 

 200, that is, when the counting extends over a volume of four thou- 

 sandths of a cubic mm. or to sixteen fields of the micrometer, and 



VOL. II. 3 R 



