ÖFVERSIGT AF K. VETENSK.-AKAD. FÖRHANDLINGAR 1899, N:0 7. 683 



m + 1 • m + 1 



Sin — ^ — X sin — 5 — /, 



???, 1 . «I + 1 



Sin — - — X sin — ^ — >t 



+ [a).((7) + '^X«)] ^ — • • • • (45) 



These expressions (44) and (45) are afterwards employed on 

 reducing the equation for Km- 



m 



Our next step is to deteriuine ^ Fm. We may write (2) 



for au arbitrary i' as follows: 



jT^j = a^{0 i' c) cos (ß — 7ny.) + la^{0 i' s) sin [ß — mx) . . . (46) 



or 



jTto = .4, cos 771X + -ß] sin 77ix (47) 



where 



A^ = aj(0 i' c) cos 6 4- aj(0 i' s) sin 6 

 B^ = a, (O i' c) sin e — a, (O i' s) cos 6 



and thus 



(48) 



^ 7^,^ = ^ [y4^ cos ?n/„ + B^ sin ?»,/,] = 



7)i = 1 TO = 1 



«I + 1 TO + 1 . m — 1 TO + 1 



sin — ^r— X cos —jr— x sin — ^ — '''^ cos —g— x 



= A. -. + Jo : 



' sin X " sin X 



. 7?i + 1 . ?B + 1 . ?ra — 1 . ?» + 1 



Sin — ^— X sm — ^^ X sin — ^ — x sin -^r— x 



+ ^ :^__, ^ — + B. ^—. — . . (49) 



' sin X " sin X 



m 



wliich is the expression sought for ^ Fm- 



m 



We have now at length to deterinine _^ S^- From the ex- 



m = l 



pression (9) we get by summation 



TO Wl 



^m — 'Sq = TC^ F,n - 1 + ^ (Cm — Cm _ 1) . . . (50) 

 TO = 1 m = \ 



and thus 



TO 7» m m m 



2 S,n = mS^ + 7r2 ^^TO-l+^ ^ (Cm — C,n-i) . . (51) 



