Empirical Proof of the Law of Error. 



337 



Mean Error ( — , our e) is \3119. Hence it follows that the 

 n 



Modulus of the Probability-curve is -55282. We are thus 



able to graduate the abscissa in aliquot parts of the Modulus, 



so as to select the proper points of comparison with each rival 



curve. Then, using the datum e=*3119 to determine the 



parameters of the rival curves, we compute the ordinates (or 



small strips of area) at each of the selected regions. 



In the accompanying table the first column contains parts 



of a second ; the second column the corresponding fractions 



Seconds. 



Modulus. 



Ex- 

 perience. 



Pro- 

 bability- 

 curve. 



Right 

 Line. 



Parabola. 



First 



Expo- 

 nential. 



Third 

 Expo- 

 nential. 



0--1... 



•181 



94 95 







130 





•1--2... 



■362 



88 89 



73 









•2--3... 



•543 



78 78 











•3-4... 



•724 



58 64 







50 



71 



•4-5... 



•904 



51 50 











•5--6... 



1-085 



36 35 





51 







■6--7... 



1-266 



26 24 



29 









of the Modulus {e.g. '1 second = 481* Modulus). The third 

 column gives for each interval the number of observations 

 actually experienced. The fourth column gives the numbers 

 as computed by Bessel for the Probability-curve. The re- 

 maining columns contain numbers computed by me for the 

 rival curves at regions selected by comparing the second 

 column of this table with the first column of the preceding 

 table. In computing the numbers I have used the same 

 approximative formula as Bessel, mutatis mutandis, viz. 



N(A'-A)[^(A) + ^(A')], 



where N is the total number of observations, here 470 ; 

 A' and A are the superior and inferior limits of an interval 

 {e.g. '1 and *2) ; and <f> is the form of a rival curve (e. g., for 



the Eight Line — 1 |, where Ci = 3e, and e=*3119). 



This empirical proof of the Probability-curve is striking, 

 and would be more striking if we extracted all the evidence 

 which the completed table would yield. If we do not care to 

 take that trouble, we are likely to obtain a better return for the 

 same labour by comparing areas rather than ordinates. This 

 comparison is effected in the accompanying table. 

 * More exactly, -18089. 



