258 



THIRD REPORT — 1833. 



limit X = h), whose base is represented by tt, then we shall 



find * 



2 , / IX f 1 r . sin 2 .r sin 3 ^ o 1 

 (p X — — [a-n + [a. — c^) b) ismx ^ \- — g &c. |- 



2 , ,s fsinZ>sinar , sin26sin2j: , sin3isin3j; ^ 1 

 + — («-«) I [2 + 2^ + 32 +&c.|. 



The trigonometrical series, in this last case, would represent a 

 series of triangles placed alternately in an inverse position with 

 respect to each other ; and a similar observation would apply 

 to the discontinuous curves which are represented by any series 

 of sines and cosines. Thus, \iy = <^ xho. the equation of the 

 curve P C C" Q, and if we suppose 



2/ = (p ^ = Oj sin X + ttg s''" 2x + a^ sin 3 x + &c., 

 between the limits and w ; and if we make A B = w, A A' 

 = 2 TT, A B' = 3 TT, &c., we sliall get a discontinuous curve, 

 consisting of a series of similar arcs, C D, a C", C D', &c., 

 placed successively in an inverse relation with respect to each 

 other upon each side of the axis of x, of which one arc C D 

 alone coincides with the primitive curve. 



If we should suppose the same curve to be expressed be- 

 tween the limits and tt by a series of cosines or 



2/ = ^ ar = «o + ^1 cos X + a^ cos 2 x + &c., 



and if we make A B = tt, A 6 = — tt, A A' = 2 tt, A B' = 3 w, 



&c., then the trigonometrical equation will represent a discon- 

 tinuous curve c? C D C D', of which the portions C D and C d, 



• Fourier has given a particular case of this series, p. 246. 



