428 THEORY OF HEAT. [CHAP. IX. 



The quantity , which denotes the limit of the second integral, 

 has an infinitely small value ; and the value of the integral is the 

 same when the limit is co and when it is oo . 



416. This assumed, take the equation 



/, , N sin p (a. x) . . 



- ^ 



, N 1 f + 



*)-] 



Having laid down the. axis of the abscissae a, construct above 

 that axis the curve ff, whose ordinate is / (a). The form of 

 this curve is entirely arbitrary; it might have ordinates existing 

 only in one or several parts of its course, all the other ordinates 

 being nothing. 



Construct also above the same axis of abscissae a curved line ss 

 whose ordinate is , z denoting the abscissa and p a very 



great positive number. The centre of this curve, or the point 

 which corresponds to the greatest ordinate p, may be placed at the 

 origin of the abscissae a, or at the end of any abscissa whatever. 

 We suppose this centre to be successively displaced, and to be 

 transferred to all points of the axis of or, towards the right, depart 

 ing from the point 0. Consider what occurs in a certain position 

 of the second curve, when the centre has arrived at the point x, 

 which terminates an abscissa x of the first curve. 



The value of x being regarded as constant, and a being the 

 only variable, the ordinate of the second curve becomes 



sin p (a oc) 

 VL X 



If then we link together the two curves, for the purpose of 

 forming a third, that is to say, if we multiply each ordinate of the 

 second, and represent the product by an ordinate of a third curve 

 drawn above the axis of a, this product is 



, , . sinp (a a?) 

 ** a x 



The whole area of the third curve, or the area included between 

 this curve and the axis of abscissae, may then be expressed by 



7 / / \ sin;? (a x) 

 J a-x 



