times can be made to depend upon another which contains the same
or
times, and so on until by continued reduction we arrive at a certain ultimate form, whose value has then to be determined.
The methods in Arbogat’s Calcul des Dérivations are peculiarly fitted for the notation and the processes of the engine. Likewise the whole of the Combinatorial Analysis, which consists first in a purely numerical calculation of indices, and secondly in the distribution and combination of the quantities according to laws prescribed by these indices.
We will terminate these Notes by following up in detail the steps through which the engine could compute the Numbers of Bernoulli, this being (in the form in which we shall deduce it) a rather complicated example of its powers. The simplest manner of computing those numbers would be from the direct expansion of
which is in fact a particular case of the development of