REPORT ON CERTAIN BRANCHES OF ANALYSIS. 263 



less m was a whole number ; for if we should denote the series 

 of cosines 



cos(m2r'if + x) + m cos {m — 2) {2r'Jf + x) + &c., by Cr, 

 and the series of sines 



sin ?n {2rit + x) + m sin {m — 2) (2rTr + x) + &c., by Sr, 

 we should find, when cos x is positive, 



' ~ cos 2 m rir ~ sin 2 m rir* 

 and when cos x is negative, 



Cr ^r 



^ ~ cos m {2r + 1) ir ~ sinm {2r + 1) ff * 



It will follow, therefore, that when r is not a whole number, p 

 will be expressible indifferently by a series of cosines or of 

 sines, unless cos 2 m r it = or sin 2 tn r tt =■ 0, when cos x is 

 positive, or cos m (2r + 1) * = 0, or sin tn {2r + l)v = 0, 

 when cos x is negative. 



In a similar manner, assuming 



X' = 1 - j^g . cos^^ + ""i % ^ ^l cos^ X 



— ^^ 1— b i a COS^ X + &C. 



1 . » . . . D 



and 



X = « •< cos X — —. — ^ — S-' cos" a:* 



3 



+ 1 . g . 3 . 4 . 5 ''"' 



.r — &c. y 



we shall find 



cos n (2 »• TT + x) = cos w(2r+ -j-Itt.X 



+ cos (?i — 1) (2 r + —\ nt . X'. 



If we suppose r to be eqvial to zero, this equation will become 



mf ,, Cn — 1) TT „, 

 cos n .r = cos -^ . X. + cos ^ . A , 



which is the form which has been erroneously assigned by La- 

 grange* and Lacroixf as generally true for all values of n. 

 Many other examples of similar tmdulating functions, ex- 



* Calcul dcs Fonctions, chap. xi. 



t Traiti' du Calcul Diff. ci Inli'g., torn. i. p. 264. 



