430 Wisconsin Academy of Sciences, Arts, and Letters. 



d (log x) 



d (log sin X) 

 d (log cos X) 



d (log tan X) 



dfx 



X 



cot X. dx 



— tan X . dx 



. dx 



sin 2x 



d (log cot x) = — 



sia zx 



. dx 



Ki] 



Furtliermore, the manner in which the errors of the number 

 a and of the arc A are connected with the errors of the Lri.!j:!]:3 

 logarithms is shown by the following equations; 



'oo- 



/(log a) =^/-(a)4-/' 



/(log sia A) = J cot A f{A)^f' 



/(log cos A) = 



1:1 



tan.4/(J)-h/' 



2r.i 



/(logtaa.D = -^-.^/u4)i-/' 

 /(logcot.4) = -^;^-j/l[^4)-h/' 



/l«) = ^. |/(loga)-/' 

 f{A)= TTtaayl . j/(!og sin .4) -/' 



= — ;— cot ^ . I /(log 003^) — /' 



= s^sin 2^. j/(logtan^)-/' 

 5.4 . [/(log cot ^)-/' 



\ 



lia 



sin 2. 



\ 



K2] 



where an / indicates tlie error accumulated in computation and 

 transferred from the logarithm to the number or arc, or from 

 the number or arc to the logarithm, and /' denotes the error of 

 a logarithm taken from the table; m = 0.43429 is the 



modulus of the Briggs system and r = 206265 is the length of 

 the radius expressed in seconds of arc. 



