any Function of Sines and Cosines. 121 



The operation by which B 3 and A 3 are eliminated is as regards 



2 ^{cos 1 + cos 3 — 2 cos2 cos 3} + 2 ^{sin 1 + sin 3 — 2 sin2 cos 3} 



= 2 2 2? 1 cos2 {cosl— cos3} + 2 2 ^i sin 2 {cos 1— cos 3}. 



The following table gives the indices for each vertical column 

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

 separately so as to eliminate all except B i} A t . 



Table XIII. 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 I. 

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

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

 stituted for the difference of two cosines. 



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

 all beneath it + . 



