307 



already deduced by Mr. Roberts from entirely different consi- 

 derations, and published in Liouville's Journal de Mathema- 

 tiques. May, 1846. 



Some other interesting results may be obtained from our 

 general formulae. Thus, if in (2) we put = 0, we will have 



from which we may deduce, by an easy transformation, 



f^^ Iog(sin0)rf0 /1\ 



and, consequently, 



r;;^^^=iio,(>).«. (8) 



If we suppose k to vanish in formulae (6) and (7), we ob- 

 tain the well-known results, originally given by Euler, 



\ log(coS(^) c?^ =z \ log (sin(^) f/0 :i:: ^tt log |. 

 »^o Jo 



Denoting v'(l — A^sin^^) by A, we can also derive from 

 the above the value of the definite integral 



For, the sum of the integrals 



C^'^ log(I + A sing) d,p ^^^^ ^°^ log (1 - A sing) 



'^0 A ' J*' A '^ 



may be found from (1), and their difference from the formula 



Jo * (l — A sin0J A V ' / 



which Mr. Roberts has demonstrated in the Journal de Ma- 

 thematiques^ May, 1846. 



2 c 2 



