120 Sir J. W. Lubbock on some Methods of developing 



Each operation moves the indices of the variable factor to the 

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

 The following table gives the indices of the sines which are intro- 

 duced by each operation when the products of two sines are substi- 

 tuted for the difference of two cosines. 



All the factors above the line which crosses the table are — , and 

 all beneath it + . 



So that, finally, for example, writing the indices of sines outside 

 the bracket, if B u and A 14 are neglected together with every B and 

 A which has a higher index than 14, 



(6) 1 =2 11 1.3.1.4.2.5.3.6.4.7.5{2? 2 sin ll-// 2 cos 11} 



(6) 2 =2 n 1.3.1.4.2.5.3.6.4.7.5{i? 2 sin 13-^ 2 cos 13} 



(6) 2 cos 11_(6) 1 cos 13=2 n .1.3.1.4.2.5.3.6.4.7.5.2.£ 2 



(6) 2 sinll-(6) 1 sinl3=2> 1 .1.3.1.4.2.5.3.6.4.7.5.2^ 2 . 



2nd Category, uneven indices. 

 2nd Operation. Elimination of B. d , A 3 * 



Let 



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

 (l)i + (1) 4 -2(1) 8 cos 3 = (2), 



(l)i + (l) 5 -2(l) 3 cos3 = (2) 2 . 



