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



If the operation is again repeated as in 



(4) 1 = (3) 1 + (3) 2 -2(2) 3 cos3, 



the constant factor which is the same throughout the same vertical 

 column is twice 



cos 10 — cos 3. 

 The following table shows the indices for each vertical column. 



Table IV.— Indices of constant factor. 



Each operation moves the indices of the variable factor to the 

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

 The second operation in the preceding example gave sin 5 sin 15, 

 sin 5 cos 1 5, the next sin 5 sin 25, sin 5 cos 25, and the fourth 

 operation sin 5 sin 35, sin 5 cos 35, 5 and 35 being the indices 

 which are found in that vertical column and in the fourth hori- 

 zontal row. 



11 9 

 cos 10— cos 1 = —2 sin -=- sin -5- 



12 8 



cos 10— cos 2= —2 sin -^- sin -~-= — 2 sin 6 sin 4 



13 7 

 cos 10— cos 3 = —2 sin -^- sin — . 



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

 troduced by each operation when the products of the two sines are 

 substituted for the difference of the two cosines. 



