any Function of Sines and Cosines. 109 



3rd Operation. Elimination of B 2 and A v 



(2)i+(2) 3 -2(2) 2 co 8 2 = (3) 1 

 (2) 2 +(2)<-2(2) 3 cos2=(3) a 

 (2) 3 +(2) 8 -2(2) 4 cos2=(3) 3 



4th Operation. Elimination of B 3 and A 3 . 



(3)j-(3) 3 -2(3) 2 cos3 = (4) 1 

 (3) 2 +(3) 4 -2(3) 3 cos3 = (4) 2 

 (3) 3 +(3) 6 -2(3) 4 cos3 = (4) 3 



Taking any vertical column as that under B l0 , the second opera- 

 tion by which B x and A x are eliminated is ; 



(l), + (l) 3 -2 (l) 2 cosl=22? 10 sin5 {sin 5+ sin 25 — 2 sin 15 cos 1} 



— 2 A l0 sin 5 {cos 5 + cos 25 — 2 cos 15 cos 1 } 

 = 2 2 2? 10 sin 5 sin 15 {cos 10— cos 1} 



— 2 2 ^4 10 sin5 cos 15 {cos 10 — cos 1} 



(1) 2 + (1) 4 — 2 (l) d cos 1 = 2 5 10 sin 5 {sin 15 + sin 35—2 sin 25 cos 1} 



— 2^i sin 5 {cosl5 + cos35 — 2 cos25cosl} 

 = 2 2 jB 10 sin5sin25 {cos 10 — cos 1} 



— 2 2 A }0 sin 5 cos 25 {cos 10— cos 1} 



(1) 3 + (1) 6 — 2 (1)4 cos 1 = 2 .B 10 sin 5 {sin 25 + sin45 — 2sin 35 cos 1} 



— 2A 10 sm5 {cos 25 + cos45 — 2cos35cosl} 

 = 2 2 jB 10 sin5sin35{coslO— cosl} 



— 2 2 A yo sin 5 cos 35 {cos 10— cos 1}. 



The factor between the brackets is the same throughout the 

 same vertical column, it may therefore be called the constant /actor. 



If this operation is repeated as in 



(3), = (2) 1 + (2)a-2(2) a cos 2, 



the constant factor which is the same throughout the same vertical 

 column is twice 



cos 10— cos 2. 



