any Function of Sines and Cosines. 113 



The table reads thus, the coefficient of B s in (4)j 



3 . 5 35 



= — 2 8 sin 1 sin - sin 2 sin - sin 4 sin ~ 



St 2 Z 



and the coefficient of A b in (4)j 



3.5 35 



as 2 8 sin 1 sin - sin 2 sin - sin 4 cos — . 



Similar expressions may be obtained when the particular values 

 are employed which the series assumes when the arbitrary angle 

 becomes successively 1, 180°+ 1, 2, 1 80° + 2, &c. This enables the 

 separation into two categories, one containing the even and the 

 other the uneven indices. 



If 



.Rj = i? -f B x cos 1 + J5 2 cos 2 + B 3 cos 3 + &c. 



+ A j sin 1 + A % sin 2 + A 3 sin 3 + &c. 

 ii' 1 = 5 + .B 1 cos(180 +l)+£ 2 cos2(180 +l) 

 + # 3 cos 3(180° + l)+&c. 

 + Ai sin (180°+ 1) + Ac, sin 2 (180°+ 1) 

 + A 3 sin 3 (l80°+l)-r&c. 

 = B —B 1 cos 1 +Bc, cos 2—B 3 cos 3 + &c. 

 — A } sin l+^ 2 sin 2 — ^4 3 sin3 + &c. 

 i (/?! + #,) = &, = # + 5 2 cos 2 + £ 4 COS 4 + &c. 

 + Ac, sin 2 + ^4 4 sin 4 + &c. 

 \ (i?! - R\) =&\ = ^ x COS 1 + #j cos 3 4- B b COS 5 + &c. 



H-^j sin 1+^3 sin 3 + ^4 5 sin 5 + &c. 



So that if R v Re,, R 3 , R 4 , &c. are the particular values of R for 

 multiples of any angle which I have assumed equal to unity, and 

 R' v .R' 2 , R' 3 , R' 4 , &c. are the particular values of R for the same 

 multiples of the angle increased by 1 80°, R x + R' v R. 2 + R'v R 3 + R' 3 , 

 &c. will contain only A { and B t where i is even, and R l —R\, 

 R^—R'c,, R 3 —R' 3 , &c. will contain only A t and B t where i is un- 

 even. 



1st Category, even indices. 



1st Operation. Elimination of B Q . 

 {\) x =\ (»i— 3fts) = Bc, (cos 2-cos 4) + B 4 (cos 4-cos 8) + &c. 

 + Act (sin 2 — sin 4) + A 4 (sin 4 — sin 8) 4- &c. 

 Phil. Mag. S. 8. Vol. 33. No. 220. Aug. 1848. I 



