any Function of Sines and Cosines. 125 



This method of mechanical quadratures is also only, when applied 

 to such series, a method of elimination, as Mr. Adams kindly 

 pointed out to me, but carried out in a different manner. 



Let 



B Q + B X cos 1 + Z? 2 cos 2 + &c. +B n -i cos (ft— 1) 



+ B n cos n + B n+i cos (ft + 1) -f &c. 



2w 

 be the series in question, and let — be substituted for 1 in the se- 

 ries, it becomes 



y, = B + B n +B 2n + &c. 



'In 

 n 



4_7T 



n 



+ &c. 



o 



+ {2* 1 + 2?n+l + 2*3n+s +&C.}C0S- 



+ {£ 2 + B n+2 + B 2 n+2 + &c.} cos 



If 



„ ,. 2 It _,2 7T __ 2 7T _,. 2j 



^Function — or F— y 2 =F2x— , Vi =Fix— , 



2y=yi+y 2 +y 3 + yn 



= ft{# + i?, 1 + £ 2M + &C.}, 



because all the other series vanish, and 



if the series converge and n is sufficiently large, 



B =-Zy. 

 In the same way it may be shown that 



^ f + ^ n _ i + ^ n+l + 5 2 „_,. + /? 2w+t .+ &c.=-Sycosi; 



and if the series converge and n is sufficiently large, 



Bi=-"Z ycosi. 

 ft 



Similarly, if the series be mixed, 



B =- 2 v, /i,= - 2 y cos i, 4=- 2 y sin i nearly. 

 n w n 



