1865.] Mr. W. H. L. Russell on the Summation of Series. 335 

 where 



F (e )=r(" + r^l>"co 8 { ('+ _J>) sine} . 



_ ge 



and \/x( x ) ' 1S su PP ose( l l ess ^an unity. 

 Hence also 



where we render the denominator rational by multiplication, and suppose 



m-2 m-3 



1 2x(#) cos md + \(x ) (x! (^) cos /w0) 2 -f- sin 2 1 

 irhere 



*(*) = 



?) is less than unity, hence XiO) is greater than unity, and therefore 

 cos m d is always positive ; hence 



The general term of the series included under the signs of definite inte- 

 gration is now of the form 



" 



belonging to a class which I have considered in my former memoir. 

 . Let us now consider the series 



and ^(ar) being rational functions of ( 



