458 Wisconsin Academy of Sciences, Arts, and Letters. 



if Lj and L2 are the tabular logarithms between which the in- 

 terpolation is to take place and « is the fraction by which the 

 difference A=^2— ^ij is to be multiplied, according to the 

 ordinary rule (the basis of which is to be considered here) 



will bo the required logarithm. In this of course « A is ac- 

 cura-ciy computed, but is expressed only in integral units of 

 the last decimal place. 



If it is assumed that/ 1 and^a are the errors of the tabular 

 logarithms Lj and Ln, it is then known that since L^ + f^ and 

 L2 + y^o ^^6 the exact values of the logarithms, the error of the 

 logariilim Li + e A will be 



/i+^(/a-/i)-^/3 



if ^3 denotes the error which arises from the curtailment of the 

 product £ A , but / is considered to be ary one of the equally 

 probable values which lie between ^^ and _;^. In each partic- 

 ular case the quantity « (which lies between and 1) will be 

 fixed and definite; nor would it be difficult to introduce into the 

 computation, if expressed nunierically, all the ;,value3 which e 

 receives during the computation. Lut since we desire to in- 

 vestigate the general case, we shall attempt to find a mean value 

 for s ; or else we may treat all the values of s as remaining 

 indefinite, and finally find a probability which will show within 

 what limits the mean error and the probable error lie. 



We shall omit the consideration of the last method although it 

 would add to the completeness and elegance of this investiga- 

 tion; but the final result which we seek would be helped but 

 little. 



So let us first inquire what values the sum 



can receive if instead of each / we write all the equally probable 



values between — 14 and + l^. The formula [6] of § 5 



serves best for this purpose. For if we substitute in it 

 ffj = 1 — £, a^ = e, a'3=l, k=3, x = \ we shall have .v=2, [s — 2m)y=l — m. 



