any Function of Sines and Cosines. 117 



Let 



(l) 1 = J B 1 {l-cosl}+B 2 {l-cos2} + .B 3 {l-cos3} 

 — A y sin \—A <2 sin 2 — A 3 sin 3 &c. 

 (1) 2 =2J, {cosl — cos2} + i? 2 {cos2 — cos 4} + -B 3 {cos 3— cos 6} 

 + ^4,{sinl— sin2} + 5 2 {sin2 — sin4}+^ 3 {sin3 — sin6} + &c. 



as before. 



If any such series be treated thus : 



(l) 1 + (l) 3 -2(l) 2 cosl = (2) 1 



B x and A x are eliminated and are not found in (2) r 

 If the quantity 



(l)j + (1) 3 — 2(1) cos i = I is formed, 

 B { and A t are eliminated and are not found in /. 



If therefore it is desired to eliminate the A and B having any 

 indices 2, 3, 4, 5, &c., similar operations must be performed with 

 cos 2, cos 3, cos 4, cos 5. 



Let 



2nd Operation. Elimination of JB 2 , A r 



(l) rt -(l) 3 -2(l) 2 cos2 = (2) 1 

 (1)«+ (1) 4 -2(1) 3 cos 2 = (2) 2 

 (l) 3 +(l) 5 -2(l) 4 cos2 = (2) 3 . 



The second operation by which A% and B 2 are eliminated is as 

 regards B v A x ; 



2 B x sin--j sin- + sin-— 2 sin -cos 2 > 



n , . If 1 5 „ 3 1 



— 2^ 1 sin-^j cos- + cos-— 2 cos -cos 2 >• 



= 2 2 ^ x sin - sin -j cosl — cos 2 J- —22^ cos -sin -jcosl-cos 2 }. 



If this operation is repeated as in 



(2) 1 + (2) 3 -2(l) 2 cos3 = (3) 1 , 



as regards B v A 1} the constant factor introduced is 



cosl— cos 3. 



The following table gives the indices for each vertical column 

 under B { , A t when the operations are conducted successively and 

 separately, so as to eliminate all except B it A t . 



