BOUNDARY PROBLEMS AND DEVELOPMENTS. 117 



Corresponding to each arc C M „ there exists another arc C^ - which is the 

 reflection of C M „ in the point X = 0. Since C~; also subtends the angle 

 co MJ , at the point X = we can write the sum (107) equally well in the 

 form 



(108) L Se («£ > + 5<p), 



But it will readily be seen that 8 (H = 8[j, for quantities which have 



the summation covering now the arc of only one (any) half of circle C 



** 

 G» 



a positive real part in a^ have naturallv a negative real part in <t-- v . 



* _* 



Hence 8\f + «W = 5 iy> and the sum (10S) i.e. (107) reduces to 



v 



jc 2tt 



It follows, therefore, that 



* 



(109) X 2= (Sjf ) - | J. 

 In consequence we have 



s2to0—**<*-o).+ «, 



i.e. 



(110) |imS2f(a?)-=iJP(a5-0)- 



/I. gfffr)- . 



The treatment of this expression is parallel to that of S^(x) • and is 

 as follows. 

 Writing 



b 



J*-= JY{x){8 f *)Z{t)R{t)F{t)-dt, 



:*)•= hS^ 



we have from (101) Sf£(x)-= — / J*-dK. 



Hence 



(HI) S%(b)-=0, 



