It will be obvious that the very same seventy-five Variable-cards may be repeated for the computation of every succeeding Number, just on the same principle as admits of the repetition of the thirty-three Variable-cards of Operations (13 ... 23) in the computation of any one Number. Thus there will be a cycle of a cycle of Variable-cards. If we now apply the notation for cycles, as explained in Note E, we may express the operations for computing the Numbers of Bernoulli in the following manner:— Again, represents the total operations for computing every number in succession, from to inclusive. In this formula we see a varying cycle of the first order, and an ordinary cycle of the second order. The latter cycle in this case includes in it the varying cycle. [Pg 65] On inspecting the ten Working-Variables of the diagram, it will be perceived, that although the value on any one of them (excepting , and ) goes through a series of changes, the office which each performs is in this calculation fixed and invariable. Thus always prepares the numerators of the factors of any ; the denominators. always receives the ()th factor of , and the ()th. always decides which of two courses the succeeding processes are to follow, by feeling for the value of through means of a subtraction; and so on; but we shall not enumerate further. It is desirable in all calculations, so to arrange the processes, that the offices performed by the Variables may be as uniform and fixed as possible. Diagram for the computation by the Engine of the Numbers of Bernoulli. See Note G. (page 67 et seq.) [Pg 66] Number of operation. Nature of operation. Variables acted upon. Variables receiving results. Indication of change in the value of any Variable. Statement of Results. Data. Working variables. 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 2 4 0 0 0 0 0 0 0 0 0 0 1 2 1 3 1 4 0 0 5 2 6 0 7 1 8 9 10 11 0 12 13 14 15 16 0 17 18 19 20 0 21 0 22 0 23 Here follows a repetition of Operations thirteen to twenty-three 24 25 by a Variable-card. 0 0 Number of operation. Result Variables. 0 0 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 [Pg 67] Supposing that it was desired not only to tabulate , , &c., but , , &c.; we have only then to appoint another series of Variables, , , &c., for receiving these latter results as they are successively produced upon . Or again, we may, instead of this, or in addition to this second series of results, wish to tabulate the value of each successive total term of the series (8), viz: , , , &c. We have then merely to multiply each with each corresponding , as produced; and to place these successive products on Result-columns appointed for the purpose. The formula (8.) is interesting in another point of view. It is one particular case of the general Integral of the following Equation of Mixed Differences:—