268 On Sines and Cosines of Multiples of a variable angle, 



cos<M±ih). A,= |s/((^^) . (...) 



It will always be convenient to make n divisible by 2, or 

 even by 4, 6, 8, &c. And thus we shall have 



cos(2w— 1) — = cos — , cos (2w — 3) 



n 



COS , &c. ; 



n 



sm (2w— 1)— = — Sin—, 

 n n 



Make 



(2»-l) 

 n 

 w — 1 )7r 



^). 



We may therefore replace (15.) by 



ztt Sztt 



sin {In •— 3) — = — sin — , &c. 

 n n 





T, 2 f 27r , 3«9r , 



J3i = — <«, COS l-WoCOS V .... + W« COS 



www 2 n 



(w— I)z7r"| 



. iTT . ^iit . in- 



sm f-i'osm f-. ••.-!- t'ra sm ^^ — 



w I ' w n 2 



DiTT 



(16.) 



The only fault of this method of finding the coefficients is 

 that it requires a large value of ^^, and consequently a large 

 number of particular values of the functions <p{<x), &c. But 

 if we give it a smaller value, the quantities 



sin(p-2)-, sin (/? + «)-, 



sin ik, sin 2/«, &c.. 



would vanish for certain values of p and i ; which would in- 

 troduce the terms Bn-i, B„, B«+j, A„_,-, A„+,, &c. ; and we 

 should have more unknown quantities than one in an equation, 

 and could not determine them without employing some other 

 means. In certain cases where the form of <p(a;), &c. is given 

 and simple, and we can ascertain with certainty the correction, 

 the method of Mechanical Quadratures, properly so called, 

 may require less labour, and may therefore be preferable. 



Gunthwaite Hall, near Barnsley, Yorkshire, 

 March 7, 1849. 



