348 Outline of general Mel/wds/ur the Development 



or if for the sake of convenience e^^" be put = z, we have 

 2 \/ — 1 sin \ = z — .-"" 



2 cos X = ~ + z . 



These trigonometric functions being now reduced to ex- 

 ponential expressions are of course givoi, and their various 

 properties may be de(hiced by mere analytical artifice. It 

 were out of place for me to waste a moment's time in the de- 

 duction of the simplest forms of their combination. I shall 

 regard them in the course of the following investigations as 

 known, and hasten to develope general methods for the disco- 

 very of the more distant and difficult formulae. The sine and 

 cosine may be combined together into series of an infinite va- 

 riety of forms; but the series whose terms involve either these 

 functions of the ascending multiples of the arc, or the ascend- 

 ing powers of the functions of the simple arc, occur most fre- 

 quently in practice, and are consequently of the highest in- 

 terest. Let us then in the first place investigate the constitu- 

 tion of series of the following forms 



A + B cos X 4- C C05 2x + D cos 3 x + E cos 4 x -f &c. 



A + B sin X + C sin 2x + D sin 3 x -f- E sin 4 x + &c. 



Happily the investigation does not present the slightest diffi- 

 culty. Eacli term of the two series is reducible to the form 



2 a/'±1 N (2" ± Z-") (3) 



and the summation of the series themselves is thus dependent 

 upon the summation of such a series as 



$ 2 = A' + B' 2 + C's' + D' z^ + &c. 

 When the coefficients A, B, C, D form either a recurring 

 progression, or are coefficients of any known transcendental 

 series, the infinite expressions may be summed at once. A 

 variety of random expressions too may be found with equi- 

 valent expressions in series. ^ z has merely to represent a 

 function of:: developeable according to rising integral powers ; 



and if either diminished or increased by <^z~'^, a slight mo- 

 dification of the arising coefficients will give their values 

 when the series correspond to given finite trigonometric quan- 

 tities. Thus the equivalence of cf: ;: to and log (1 + z) 

 gives us the seven series 



I = C05 X — C05 2 X + cos 3 X Sic. 

 — i = cos-K + COS 2 X -{- cos 3 X &c. 



i tan — = sin x — sin 2 x H- s/n 3 x &c. 



^ cot 



