log2. 



M^ & = | log 3. 



192 MATHEMATICAL NOTE. 



But/(0)=0, since log 1=0. Therefore 



| 



Therefore 



The singularity of this method, and its applicability in other 

 cases give it interest : but, as the writer of the paper already 

 noticed pointed out to me, the integral may be got by assum- 

 ing x = tan y ; it then becomes 



IT 



I log(l+tany)<Zy, 



^0 



and, by his fundamental equation, 



IT 7T 



4 log (1 + tan y} dy = J* log jl + tan (~ - ^J dy : 



tan y 2 



a n I - 2 , = + rn ^ = rn _, 



and therefore 



n 



2[*log(l+tan#)%= j log 2, 



^0 



whence the truth of M. Bertrand's result is obvious. 



