258 



THIRD REPORT — 1833. 



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

 find * 



2 f 



^ jr = — {« TT + (a — a') 6} \ 



2 , ,. [si 



sin 2x sin 3 



+ 



sinfisina: ?,m2bs\n2x , sin36sin3a: ^ 



22 ' 32 ~ ^ "'"•J- 



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. Tlius, if y = f ,r be the equation of the 

 curve P C C" Q, and if we suppose 



7/ = (p a: = «j sin .r + «2 ^^'^ 2x + a^ sin S x ■{- &c., 

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

 = 2 TT, AB' = 3 7r, &c., we shall 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 



y = <p X =: Oq + «! cos X + Uc^ COS 2 X + &C., 



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

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

 tinuous curve dCD O D', of which the portions C D and C d, 



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



