334 Mr. W. H. L. Russell on the Summation of Series. [June 15, 

 fm+-11 f -^ 



1 M+W / 



(where log log e - is negative), 

 i 



+ f ' du e^s 



Jo 



(where log log t - is positive). 



By means of these transformations the series is reduced to forms consi- 

 dered in my previous investigations ; for the general term of the trans- 

 formed series included under the signs of definite integration is of the form 



Pa* m+ {cos h(m+nz)- sin 



a form which I have discussed in the memoir in the Philosophical Trans- 

 actions mentioned at the beginning of this paper. 

 Next let us investigate the series 



where \^(x) and \(x) are identical functions of (x). 

 We transform as follows : 



COS P> 



remembering that log^ u is negative, 



1-2V / X( J? ) COS0 + 



