any Function of Sines and Cosifies. 123 



Let 



R l,l=4 (^1,1 + 2*1,2+ #1,3 + ^1,4) 



R i,2 = 4 (^1,1 + ^1,2—^1,3—^1,4) 

 R i,3 = 4 (^1,1—^1,9 + ^1,3— #1,4) 



R i,4=4 (^1,1 — ^1,2—^1,3 + ^1,4) 



H hl =z B + B cos 2 + i? 4 cos4 + &c. 



R 1}1 contains only B 0i # 2 , B 4 , &c, having even indices. 



R 1>2 = B x cos 1 + Z? 3 cos 3 + B s cos 5 + &c. 



R 1>2 contains only B v B s , B b , &c, having uneven indices. 



R 1)3 = A^ sin 2 + A 4 sin 4 + ^4 6 sin 6 + &c. 



R 13 contains only A^ A 4 , A 6 , &c, having even indices. 



R M =^! sin 1-f ^ 3 sin3+^4 5 sin5 + &c. 



R 1>4 contains only A v A 3 , A 5 , &c, having uneven indices. 



Similarly, if 



R ifl denote the value of the series R clue to the angle i, 



-"t,2 • h 



B i3 180° + *, 



i?,- 4 -180° + t, 



R il = B -\- B x cos i + 5 2 cos2i + &c.4^i sini + .<4 2 sin 2i-f&c. 

 B ifi =zB + B x cosi + Z? 2 cos2 i + ikc — A x sini— ^ 2 sin2i + &c. 

 B i3 —B —B l cos« + J5 2 cos2 i + &c— A x sin i + A% sin 2 i + &c. 

 R i4 = B — jBj cos « + j3 2 cos 2 i--)-&c. + ^4j sin i—-^ sin 2i+&c. 



and if 



R i, 1 = 4 0^, 1 + ^«,2 + R i,3 + R i,4) 

 R *,2 = J ( fy, 1 + ^*,2 — ^i,3 — ^',4) 



