Bremihcr — Errors Affecting Logarithmic C amputations. 443 

 80 in these twenty sums selected at random there aro 



13 errors between and 1 

 4 " " 1 " 2 



2 *« «* 2 " 3 



1 ** •« 3 " 4 



We should compare these numbers wi.h those found afoove, 

 dividing of course by 500 since in this case there are only twenty 

 sums, instead of 10,000. After this division the probability is 



11,2 errors between and 1 

 64 *» it 1 " 2 



2,0 *• »* 2 " 3 



04 «« «« 3 ♦* 4 



SO that there is as great an agreement as could be expected in so 

 small a number of sums. 



The smallness of the probability of the larger errors is seen 

 both from this result and from tine example above, since in 

 10,000 errors of this kind, only one exceeds five units; the 

 extreme limit of ten units is so far remote from the range of er- 

 rors which actually occur that errors of that sort are to be re- 

 garded as practically non-existent. For by actual computation 

 the probability of an error between 9 and 10 is found to be 1 : 

 2432 900 000 000 000 000. Moreover the extreme limJt of er- 

 ror is of little value in logarithmic computations, as will appear 

 below, where more accurate criteria are proposed for testing such 

 computation. 



§ 7. 



In discussions of the probability of errors the probable error 

 and the mean error must receive especial consideration. The 

 limits within which exactly half of the total number of errors is 

 situated is called the probable error. Thus the probable error, 

 according to our formula, is (s — 2 m) r ii m satisfies the equa- 

 tion 2—2 Wm =^ or Wvi =i. This equation cannot be solved by any 

 direct method. 



The mean error denotes the arithmetical mean of all possible 



