114' Sir J. W. Lubbock on some Methods of developing 



(l) 2 =I(ft 2 -ft 3 ) = £ 2 (cos4-cos6) + £ 4 (cos8-cosl2) + &c. 



+ At (sin 4 - sin 6) + A 4 (sin 8 - sin 1 2) + &c. 



The indices are given in the alternate columns of Table II., and 

 the indices of the arguments which are introduced when the differ- 

 ences of the sines and cosines are replaced by products are given in 

 the alternate columns of Table III. 



Let 



2nd Operation. Elimination of B 2 and A 2 . 



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

 (l) 2 +(l) 4 -2(2) 2 cos2 = (2) 2 

 (l) 3 +(l) 4 -2(2) s cos2=(2) 3 



3rd Operation. Elimination of J3 4 and A 4 . 



(2) 1 + (2) 3 -2(2) 2 cos4=(3) 1 

 (2) 2 +(2) 4 -2(2) 3 cos4=(3) 2 

 (2) 3 + (2) 5 -2(2) 4 cos4 = (3) 3 



Taking any vertical column as that under B 10 , the second opera- 

 tion by which B^ and A 2 are eliminated is, 



(1), +(1) 3 — 2(l) 2 cos 2 = 2 B 10 sin 5 {sin 15 + sin 35 — 2 sin 25 cos 2} 



— 2^4 10 sin 5 {cos 15+cos35 — 2cos25 cos2} 

 = 2 B 10 sin 5 sin 25 {cos 10 — cos 2} 



— 2 A 10 sin 5 cos 25 {cos 10 — cos 2}. 



The indices of the constant factors are given in the alternate co- 

 lumns and the alternate horizontal rows of Table IV., and the in- 

 dices of the sines which are introduced when the products of sines 

 are substituted for the difference of the cosines, are given in the al- 

 ternate columns and the alternate horizontal rows of Table V. Each 

 operation as before moves the indices of the variable factor to the 

 numbers which are in the next horizontal row beneath in Table III. 



If 



2 sin 1 be represented by i 



2 2 sin 1 sin 2 be represented by ... 2 



2 3 sin 1 sin 2 sin 3 be represented by 3, 

 &c. 



