196 UNIVERSITY OF VIRGINIA PUBLICATIONS 



We assume all numbers to be real and / {z) to be a continuous function 

 in the interval — ir to + x inclusive of these numbers. 



Substitute the value of x in (15) in the integrals in (14). In the firsb 

 integral in (14) make the substitution t for (2?i + 1)2, this integral then 

 becomes 



./ o \ ' ' sm 



2n + l 



Since t lies between and c < tt the function ^^^ is always positive and 



this integral, by the first law of the mean value, can be written equal to 



J^;(_2fc-f)-\psin_.,^_ 



2?i+l 



where f is some number between and c. Hence when n is infinite this 

 integral has the value 



/(7r-0)Jj'^rf^ (18) 



Making the same substitutions in the second integral of (14), it can be 

 decomposed into the following 



Uo Jo J^-JTi+i 



The first of the integrals on the right of (19) can be written 



2 



of which the first is well kno^^^l to have the value 



The transformation z = ir — t turns the second integral of (20) into 



