390 



PRACTICAL MATHEMATICS 



Let the base line of such a curve be divided into m equal parts 

 and the ordinates y , y lt ?/ 2 , . . . y m drawn to the curve at each 

 point of division. The lengths of these ordinates will give the 



values of y when has the values 0, , , . . . 2-K, re- 



771 Tfv 7TL 



spectively. 



FIG. 126. 



(1) 



B n 



T- H/ 



2TCJ 



B is evidently the average ordinate of the curve obtained by 

 plotting horizontally and y vertically, and therefore 



B = (sum of the ordinates) 



2_ 



m 



The ordinate y m is not to be included, since it forms the initial 

 ordinate of the next period. 



(2) Now 



= - y cos 



B n is evidently twice the average ordinate of the curve obtained 

 by plotting horizontally and y cos nft vertically. 



Then B n = {sum of the ordinates of the 0, y cos n0 curve} 



2 



{il n cosO"+?/ 1 cos \-y<> cos K . . ?/_i cos - 



m 



+1J-, COS -- (-W, COS 



m m 



2n(m - I)TC! 



i COS - * - r 



This will give the coefficient of any cosine term in the resulting 

 series by giving n the required value. 



