Braniker — Errors Affecting Logarithmic Computations. 457 

 will be the probability of an error within the limits ± C 



^P-\/ -^ will be the probable error, 



and <^n / -^ the mean error, 



expressed in terms of the last decimal place. Or, if there 

 be written in pla,ce of ^^^ its value a^^-[-az'^-\- — +a^ =^rt2 



3a 8 



V 'Q 



will be the probable error, 



and 1 / ^ - the mean error. 



§ 12. 



Before we nse these formulas, let ns examine more carefully 

 the general nature of logarithmic computations. First, it must 

 not be assumed that every time we pass over from logarithm to 

 number or from number to logarithm an error arises whose 

 limit is the half of unity, since we may have added to the 

 tabular logarithm a proportional part. Also, the probability of 

 error in different intervals is not the same, since many sources 

 of error exist. Lastly, the limits and the probabilities of error 

 vary according as the transition is made from number to log- 

 arithm or from logarithm to number. Hence it will be neces- 

 sary to modify equations [1] and [2] of § 2, in which it was as- 

 sumed that there was only a single source of error arising with 

 each single transition. Therefore in order that everything may 

 be correctly expressed by the formulas, let us more carefully con- 

 sider the transition from number to logarithm. 



If it is required to find the logarithm corresponding to a cer- 

 tain number, it is commonly done by adding to the tabular 

 logarithm a proportional part from the table of differences. So 



