460 Wisconsin Academy of Sciences^ Aris, and Letters, 

 iaid tlie j^robable error and the mean error are found to bo 



for s = 0,0 0,1 0,2 0,3 0,4 0,5 



probablo error = 0,293 0.279 270 0,2G3 0,232 0,261 

 mean error = 333 318 307 298 293 0,292 



'Hence it appears that it mattci*s little what value 8 has, since 

 the values which the mean error and the probable error receive 

 for various values of e differ among themselves but little. In 

 order to adopt a single average value and thus arrive at a more 

 convenient form for computation, let U3 put « = i , for v/hich 

 the probable error will be 0.270 and the mean error 0.307. 



Therefore we may put the error of a logarithm found by in- 

 terpolation as 



/ = ^A-f i/2+/*3 [20] 



If the trn-pciition is from logarithm to number or arc, the er- 

 ror which is to be assumed as occurring in the logarithm has the 

 form 



— (1— O/l-f/o. 



For there are given, the logarithm L = log (p{a) , the tabular 

 logarithms L^ and L2 between which the interpolation is to be 

 performed, the difference Zo — L^ = ^ and Z, — L^ = S. 

 In order that the argument a corresponding to the logarithm L 



may be found, the fraction -r is compAited and added to the 



argument of the logarithm L^. Thus if Li + /i, L^ + /z, are 

 the true values of the logarithms, and therefore 



5— /j. and ^-j-fz—fx 

 are the true differences, we have 





