684 



SCHULTZ-STEINHEIL, THE ARGUMENT X™. 



m 



^ S„)=/n*So + _^ 



m = l 



as 



^ ^ -^71 + ^ {Cm — Cm — l) 



m = 1 /)! = 1 



—TtJ^Fm + m7TrQ (52) 



?7i = 1 /)t = 1 



If we consider (44) and (49) we get 



TO m 

 7t ^ Fm + 2li {Cm. C,n-i) — 



D 



i(i) 



sm 



m = l 

 + 1 



m + 1 

 X, COS — Ti — >t 



1 m + \ 



- Y. COS — -^ — 



<1) 



sm X 



. + 1 . m + 1 



Sin — H — z sin — ^ — X 



+ i>: 



,(2) 



+ E. 



where 



sm z 



+ ^; 



sin — 9^ 



(2) ^ 



Sin X 



y. sm 



+ 



+ 1 



sin z 



Df=7tA, + 0,(u) + -%{g) I 



E'-^^ = 7tB._ + 0o(i7) + % (n) .) 

 We raay write (54) as foilows: 



■))(. 7)( 



^^ -Tto + ^ (6„i — Cm _ i) = 



7)1 = 1 7)1 = 1 



^(1) ^(2; 



= + r. • sin (m + 1) /. + TT-T-^^ — fsin mrA — sin zl + 



2 sin z 



2 sin z 



^ 



(1) 



^(2) 



(53) 



(54) 



(55) 



+ --7^^ — ri — cos (m + 1) zl + ^r-^ — [cos z — cos mz 1 . (56) 



2 sin z 



2 sin z 



From (56) we get by sum mation with respect to m 



m l VI m 



^ 'j TT ^ Fm + ^ {Cm Cm _ i) 



OT = 1 ( 7)1 = 1 



771=1 



r, ?n + 1 j^ m — 1 ^ I cos?nz ^ sin 7?rz] . 



^^•^- + ^^'^- + 2r^^hrz7 + ^^imz7 • ^^^> 



m = l 



