Bremiker — Errors Affecting Logarithmic Computations. 447 



But the function which is to be integrated here becomes a 

 maximum if x is taken equal to zero. As x increases, the num- 

 erator repeatedly becomes zero, while the denominator continu- 

 ally increases, so that we have a sufficiently exact value for the 

 integral if we take large finite quantities — g and + g \n place 



of the infinite limits. By this substitution sin ^— is changed 



2ii 



into ^ for the quantity n is assumed infinite. Thus after 



2/?. ^ '' 



division bv 



4/ 



(2Ai)^ai«:,....a 



we have the probability Vq of an error between the limits — e 

 and + c. 



1 i t^in fTj X sir. a^ x sin ex x sin ex 



^ ■ a^x a^x oi X x 



Then if each sine is expressed as a scries, so that 



and similarly for 



a^x 



we havo 

 log 



.o.,-^^y.....o.(!!^) 



a^x a^x a x / 



if we put 

 and 



