462 Wisconsin Academy of Sciences, Arts, and Letters. 



Accordingly the formulas [1] and [2] of § 2 are slightly 

 modified, thus: 



/(log a) =rif^a)^f' 



f (log sin ^) = -cot A.f{A)-\-f 



"71 



f {log COS A) = ——tan A. f{A)+f 

 f(logtan^)=^_^-./(^)-f/' 

 / (log cot A, = - '-^^r^^ . /(^)+/' 

 /(«) = ~ \ /(log a) +r \ 

 f{A) = ^tan.l |/(log sin A)^r} 

 = - ^cot A I /(log cos A)-\-f' } 

 = — sin A cos A \ f (log tan A) -}- f t 



lib ^ J 



= sin A cos A 



m 



|/(!cgcot^)+/'| 



PnrtLermoro in tlie transition from log tan A and Icri^ cot Al to 

 log sin A and leg cos A the following for-niihis arc used: 



/(log sin A) = cos^ A ./(log tan A) -\- co3^4 .f"-\-f' 

 /(log cos ^) = — 8in« A ./(logtan^l)- sin-^ ./"A^f 

 /(log sin A) = — cos2 ^ . / (log cot ^1) — cos-'.l ./"+/' 

 /(log cos ^) = sin2 A . /(log cot ^) -f sinM . / "+/" 

 /(log sin ^1) = — cot« -4 . / (log cos J.) — cot^J^ . f'-if 



"whpre /' and /" are to be taken according to formulas [20] and 

 1[21]. 



in which all the quantities fj, fz etc., are entirely independent, 

 BO that each one denotes any one of the equally probable values 

 between — % and + %, whose number is infinite. 



