REPORT ON CERTAIN BRANCHES OF ANALYSIS. 257 



The hypothesis of n being infinite would convert the series 

 for «„ into the definite integral * 



2 /»0 



2 /»0 . , 



— / ax sin mx ax. 



COS a; 



if we make r = x and r = d x : or otherwise if we 



?^ + 1 w + 1 



assume the existence of the series 



(p X ■=■ a^ sin a: + «2 sin 2 x + . . . a,n sin m « + &c., 



it may be readily shown, by multiplying both sides of the equa- 

 tion by sin m x d x, that 



a,n = — / (p X sin m X d X : 



and in a similar manner, if we should assume 



<p X = Oq cos .r + «i cos x + . . . a^ cos 7n x -\- &c., 

 that 



cr^ = — / <p X cos mx a X -f . 



Thus, if we should suppose ip jp = cos x, we should find 



4 C 2 4 . 6 1 



= — \ Tj — ^c sin 2 ^ + H — p sin 4 x -f -p — = sin 6 a; + &c. I • 

 »r[l.o o.o o./ J' 



a very singular result, which is of course only true between the 

 limits and it, excluding those limits %. 



If we should suppose (p .r = a constant quantity — between 



the limits and a, and that it is equal to zero between a. and tt, 

 we should find 



(1 — cos«) . (1 — cos2«) . _ (1 — cosoa) 

 (p X = ^ ^ sin X + ^ ^ -' sm 2 ^ + ■ ^ ~^ x 



sin 3 X + &c., 

 excluding the limiting value u, when the value of the series is 



only -^§. 



If we should suppose <p x =. 'Dj" . a-x + "D,^ . («' x + /3'), 

 which is the equation of the sides of a triangle (excluding the 



• Poisson, Journal dc I'Ecole Polytcchniqve, cahier xix. p. 447. 

 + Fourier, Th/'oric de la Chaleiir, pp. 235 & 240. 



\ IhUL, p. 233 ; Poisson, Joimial de I'Ecole Polytechmque, cahier xix. p. 418. 

 § Fourier, Tlicorie de la Chaleur, p. 244". 

 loutj. s 



