SECT. IV.] EXAMINATION OF AN INTEGRAL. 439 



In equation (A), namely, 



we can replace the sum of the terms arranged under the 

 sign 2 by its value, which is derived from known theorems. 

 We have seen different examples of this calculation previously, 

 Section III., Chap. in. It gives as the result if we suppose, 

 in order to simplify the expression, 2?r = X, and denote a-# 

 by r, 



_-+.; . . . sin r 



2j cos ir = cos ?r+ sin ir - -. 

 -j J versmr 



We must then multiply the second member of this equation 

 by cZx/(a), suppose the number j infinite, and integrate from 

 a = - TT to a = + TT. The curved line, whose abscissa is a and 

 ordinate cos^V, being conjoined with the line whose abscissa is 

 a. and ordinate /(a), that is to say, when the corresponding 

 ordinates are multiplied together, it is evident that the area of 

 the curve produced, taken between any limits, becomes nothing 

 when the number j increases without limit. Thus the first term 

 cosjr gives a nul result. 



The same would be the case with the term sinjr, if it were 

 not multiplied by the factor - ^ ; but on comparing the 

 three curves which have a common abscissa a, and as ordinates 



sm r 

 versin r 



sin ? 



, we see clearly that the integral 







c/a/(a) sinjV 



versiii r 



has no actual values except for certain intervals infinitely small, 



namely, when the ordinate - becomes infinite. This will 



versin ?* 



take place if r or a x is nothing ; and in the interval in which 

 a differs infinitely little from x, the value of /(a) coincides with 

 f(x). Hence the integral becomes 



J 



r sin J r &amp;gt; or 4/(.r) j ~ sin jr, 



