We have now explained how the engine might compute (1.) in the most direct manner, supposing we knew nothing about the general term of the resulting series. But the engine would in reality set to work very differently, whenever (as in this case) we do know the law for the general term.
The two first terms of (1.) are
and the general term for all after these is
which is the coefficient of the (
term. The engine would calculate the two first terms by means of a separate set of suitable Operation-cards, and would then need another set for the third term; which last set of Operation-cards would calculate all the succeeding terms ad infinitum; merely requiring certain new Variable-cards for each term to direct the operations to act on the proper columns. The following would be the successive sets of operations for computing the coefficients of
terms—