62 Mr. H. M. Macdonald. The Bendiiig of [M ;iy 12, 



that is 



cos x 

 Writing 



the above becomes 



8 = Si + S 2 + S 3 + S 4 - 85 - $r, - 87 - Ss, 

 where 





S 4 = S/, (n)e-'->- S 8 = Z/iOO. 



o *o 



If any value of Wj of ?? be chosen, and n is written i + r, the 

 quantities Xn> <#>n can be expressed in the form A + A! - + A-2 2 + . . . 



for values of v for which the series converge ; similarly f or /i (),/(*). 

 The group of terms in any of the series Si etc., for which this holds for 

 MI, will contribute nothing to the result unless the coefficient of v in 

 the exponent vanishes ; also when the coefficient of v in the exponent 

 vanishes and the coefficient of v 2 does not, the sum of the corresponding 

 group of terms 





the upper or lower sign being taken according as B 2 is positive or 

 negative.* 



* Lorenz, ' (Euvres Scientifiques/ vol. 1, p. 425. 



