Brcmilccr — Errors Affecting Logarithmic Compuiations. 441 



§ 6. 



To show by an example how the formula is to be used, let us 

 take twenty logarithms from the tables and form their sum. All 

 the logarithms taken from the table are affected by an error, of 

 which the limit is one-half the last decimal place; it is required 

 to find the error of the sum. 



Here we take a^^a^^ =a^ =1, ^=20 and y = y^ oi the last 



decimal place. The sums s^a, of which there 20, each equal 1 ; 

 «2«'=2, there being -^^-^ of these sums; S;^a=3, and of these there 



are 



20 . 19 ■ 18 

 1.2.3 



; etc. So Wm becomes according to [6] 



2^ ) m^°-20(m-l)^«+^^(^-2)^°- -?^- ^l •3^^ (m-3)»° 



+gg-l^-lg'lVz-4)^o-.. 

 1.2.3.4 



The extreme limits of the error are — 10 and + 10, and the prob- 

 ability that the error (without regard to sign) is between and 

 10 — m is 1 — 217771. Then if instead of m we put in turn 1, 

 2, 3, . . .9, we have the probability of an error between the 

 limits and 9, 8, 7, . . . 1. Performing the subtraction and 

 putting the total number of different errors equal to 10,000, we 

 have 



that is to say, among 10,000 sums which are made by the addi- 

 tion of 20 logarithms taken from the table, 5,586 are affected by 

 an error between the limits and 1 in terms of the last place of 

 decimals, and the error of no sum is greater than 6. 



This computation can be tested to some extent by several trial 

 additions of twenty logarithms. Take as an example the sum 



