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 [Pg 59] which is in fact a particular case of the development of mentioned in Note E. Or again, we might compute them from the well-known form or from the form or from many others. As however our object is not simplicity or facility of computation, but the illustration of the powers of the engine, we prefer selecting the formula below, marked (8.). This is derived in the following manner:— If in the equation (in which , ..., &c. are the Numbers of Bernoulli), we expand the denominator of the first side in powers of , and then divide both numerator and denominator by , we shall derive If this latter multiplication be actually performed, we shall have a series of the general form in which we see, first, that all the coefficients of the powers of are severally equal to zero; and secondly, that the general form for the coefficient of the 2()th term (that is of even any power of ), is the following:— [Pg 60] Multiplying every term by () we have which it may be convenient to write under the general form:— , , &c. being those functions of which respectively belong to , , &c. We might have derived a form nearly similar to (8.), from the coefficient of any odd power of in (6.); but the general form is a little different for the coefficients of the odd powers, and not quite so convenient. On examining (7.) and (8.), we perceive that, when these formulæ are isolated from (6.) whence they are derived, and considered in themselves separately and independently, may be any whole number whatever; although when (7.) occurs as one of the ’s in (6.), it is obvious that is then not arbitrary, but is always a certain function of the distance of that from the beginning. If that distance be = , then