100 A. Mukliopadlij'ay — On Foisson's Integral. [No. 1, 



III. — On Poissons Integral. 

 By AsuTosH MuKHOPADHTAY, M. A., F. E,. A. S., F. R. S. E. 



[Eeceived January 16th ; — Bead February 1st, 1888.] 



Contents.. 

 §. 1. Introduction. 

 §. 2. Transformations of the integral. 

 §. 3. Symbolic value for tt. 

 §. 4. Geometric interpretation. 



§ . 1 . Introduction. 

 The definite integral 



sin^^ X dx 

 o (l-2acosa; + a«)'' ^^^ 



has been often discussed by mathematicians ; it seems to have been first 

 considered by Poisson in his memoir Suite du Memoire sur les Integrates 

 Definies published in the Journal de VEcole Polytechnique (1815), t. X, 

 Cah. XVII, p. 614. Poisson first attacks the more general integral 



sin^ X dx 



(^) 



£ 



'=fo^ 



2a cos ^4-^2)"' 



Differentiating with regard to a, and integrating by parts, he obtains 

 the differential equation 



which is satisfied by the value of the definite integral in question. As 

 an integrable case of this equation, Poisson makes the coefficient of y 

 nugatory, by putting p = 2n; so that the integral (2) assumes the form 

 (1), while the subsidiary differential equation becomes 



the solution of which is 



d^^2n^ dy^^ 

 da^ a da ' 



= Ci-l-C2a-2^ 



Poisson next proves that by virtue of the general nature of the inteo-ral 

 expression, we must have Cg = 0, while, by making a = 0, it easily follows 

 that 



-I 



TT 



sin2^ cc dx. 



Hence, it is finally shewn that 



I -— I sm"xdx (R\ 



