SECT. VI.] GENERAL FORMULA. 185 



If n is an integer, and the value of x is equal to TT, we have 

 s = n \(f&amp;gt; (x) sinnxdx. The sign + must be chosen when n is 



odd, and the sign when that number is even. We must make 

 x equal to the semi-circumference TT, after the integration in 

 dicated; the result may be verified by developing the term 



| (/&amp;gt; (x) sin nx dx, by means of integration by parts, remarking 



that the function &amp;lt; (x) contains only odd powers of the vari 

 able x, and taking the integral from x = to x = TT. 

 We conclude at once that the term is equal to 



o 



If we substitute this value of - in equation (B), taking the 

 sign + when the term of this equation is of odd order, and the 

 sign when n is even, we shall have in general I $(x) sin nxdx 



for the coefficient of sin?z#; in this manner we arrive at a very \ 

 remarkable result expressed by the following equation : 



7T(j&amp;gt;(x) = since I sin x$(x) dx + sin 2x /sin 2#&amp;lt; (x) dx+&c. 

 J J 



in/ic lsini#&amp;lt; (x) dx + &c .............. f. (D), / 



. 



&quot;sX 



the second member will always give the development required 

 for the function &amp;lt;/&amp;gt;(#), if we integrate from x = to # = 7r. 1 



1 Lagrange had already shewn (Miscellanea Taurinensia, Tom. in., 1760, 

 pp. 260 1) that the function y given by the equation 



y = 2 (iTV, sin X r -rr AX) sin xir + 2 (5TV r sin 2X r Tr AX) sin 2xir 



r=l r=l 



+ 2 (iT Y r sin 3X r 7r AX) sin 3xir + . . . + 2 (S^Y r sin nX r v AX ) sin nxir 

 receives the values F 1} Y^, Y 3 ...Y n corresponding to the values X lt X 2 , X 3 ...X n of 

 x, where X r = , and AX . 



Lagrange however abstained from the transition from this summation-formula 

 to the integration-formula given by Fourier. 



Cf. Riemann s Gcsammclte Mathcmatische Werke, Leipzig, 1876, pp. 218220 

 of his historical criticism, Ucber die Darstellbarkeit einer Function durch eine 

 Trigonomctritche Reihe. [A. F.] 



