REPORT ON CEKTAIN BRANCHES OF ANALYSIS. 2G3 



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

 of cosines 



cos {m2rir + x) + w cos (m — 2) (2 r * + .r) + &c., by Cr, 

 and the series of sines 



sin m {2rTf + X) + m sin {tn — 2) (2rTr + x) + &c., by Sr, 

 we should find, when cos x is positive, 



Cr ^r 



^ "" COS 2 m rir ~ sin 2 m rv' 

 and when cos x is negative. 



and 



^ cos m {2 r + \) ir sin w (2 r + 1) ** 



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 mrie = or s'm 2m r if =■ 0, when cos x is 

 positive, or cos m (2 r + 1) * = 0, or sin ?n {2r + 1) w = 0, 

 when cos x is negative. 



In a similar manner, assuming 



1 . 2 ... 6 



r («2-P) 3 



X = n < cos X — ^:j — g — ^ cos'^ j:* 



we shall find 



cos 7« (2 »• TT + x) = cos n ( 2 r + Y ) '^ • ^ 



+ cos (w - 1) (2 r + 4") 'T . X'. 



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



nif ^ (n — 1)* ^, 

 cos n X = cos -^ . X + cos 5 . A', 



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

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

 Many other examples of similar undulating functions, ex- 



* Cakttl den Funciio)'s, chap. xi. 



t TVaite du Caktil Diff. ct lu/erj., torn. i. p. 261. 



