RF.PORT ON CERTAIN BRANCHES OF ANALYSIS. SijT 



The liypotliesis of n being infinite would convert the series 

 for a,n into the definite integral * 



2 ro 



2 ro . , 



— / <^ X sm mx ax, 



cos 



if we make rr = x and — ; — r = d x : or otherwise if we 



71+1 71 + [ 



assume the existence of the series 



<f) X = Oy sin X + «2 sin 2 x + . . . Om sin 7n 7i + &c., 



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

 tion by sin m x d x, that 



a,n "= — I <p X sin m x d x: 



and in a similar manner, if we should assume 



<tix ■=. Qq cos o; + «i cos X + . . . a^ cos ?» .r + &c., 

 that 



«OT = — / 1^ X cos m X d x-\. 



Thus, if we should suppose <f x — cos x, we should find 



4r2.^ 4.. Q . n ol 



X = — \ rj — ^ sin2x + 3 — ? sin 4 x •{- -^ — ;^ sm 6 a; + &c. i- • 

 ?r[1.3 o.o 5.7 J' 



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

 limits and tt, excluding those limits %. 



If we should suppose (p x =■ a constant quantity -^ between 



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

 we should find 



(1 — cosa) . (1 — cos2«) . _ ■ (1 — cos 3 a) 

 <p X = ^- — -g sm X + ^ -' sm 2x -{- ^ J x 



sin 3 X -{- &c., 



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



only ^§. 



If we should suppose <f x = 'Dj" . xx + ■"'DiT . («' x + j3'), 

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



* Poisson, Journal de I'Ecole Polytecliniqiie, cahier xix. p. 447. 



t Fourier, Thi'oric de la Chaleur, pp. 235 & 240. 



+ Ibid., p. 23S ; ¥o\s,so\\. Journal de l' Ecole Polytechniqiie, cahier xis.. p. 418. 



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



1833. s 



