In this state the only remaining processes are first: to transfer the value which is on V_13, to {V}_{24}; and secondly to reduce V_6, V_7, V_13 to zero, and to add[30] one to V_3, in order that the engine may be ready to commence computing B_9. Operations 24 and 25 accomplish these purposes. It may be thought anomalous that Operation 25 is represented as leaving the upper index of V_3 still = unity. But it must be remembered that these indices always begin anew for a separate calculation, and that Operation 25 places upon V_3, the first value for the new calculation.

It should be remarked, that when the group (13 ... 23) is repeated, changes occur in some of the upper indices during the course of the repetition: for example, ^3V_6, would become ^4V_6, and ^5V_6.

We thus see that when n=1, nine Operation-cards are used; that when n=2, fourteen Operation-cards are used; and that when n>2, twenty-five Operation-cards are used; but that no more are needed, however great n may be; and not only this, but that these same twenty-five cards suffice for the successive computation of all the Numbers from B_1, to {B}_{2n - 1}, inclusive. With respect to the number of Variable-cards, it will be remembered, from the explanations in previous Notes, that an average of three such cards to each operation (not however to each Operation-card) is the estimate. According to this the computation of B_1 will require twenty-seven Variable-cards; B_3 forty-two such cards; B_5 seventy-five; and for every succeeding B after B_5, there would be thirty-three additional Variable-cards (since each repetition of the group (13 ... 23) adds eleven to the number of operations required for computing the previous B). But we must now explain, that whenever there is a cycle of operations, and if these merely require to be supplied with numbers from the same pairs of columns and likewise each operation to place its result on the same column for every repetition of the whole group, the process then admits of a cycle of Variable-cards for effecting its purposes. There is obviously much more symmetry and simplicity in the arrangements, when cases do admit of repeating the Variable as well as the Operation-cards. Our present example is of this nature. The only exception to a perfect identity in all the processes and columns used, for every repetition of Operations (13 ... 23) is, that Operation 21 always requires one of its factors from a new column, and Operation 24 always puts its result on a new column. But as these variations follow the same law at each repetition, (Operation 21 always requiring its factor from a column one in advance of that which it used the previous time, and Operation 24 always putting its result on the column one in advance of that which received the previous result), they are easily provided for in arranging the recurring group (or cycle) of Variable-cards.

We may here remark that the average estimate of three Variable-cards coming into use to each operation, is not to be taken as an absolutely and literally correct amount for all cases and circumstances. Many special circumstances, either in the nature of a problem, or in the arrangements of the engine under certain contingencies, influence and modify this average to a greater or less extent. But it is a very safe and correct general rule to go upon. In the preceding case it will give us seventy-five Variable-cards as the total number which will be necessary for computing any B after B_3. This is very nearly the precise amount really used, but we cannot here enter into the minutiæ of the few particular circumstances which occur in this example (as indeed at some one stage or other of probably most computations) to modify slightly this number.

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:— array of equations Again, array of equation represents the total operations for computing every number in succession, from B_1 to {B}_{2n-1} 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.

On inspecting the ten Working-Variables of the diagram, it will be perceived, that although the value on any one of them (excepting V_4, and V_5) goes through a series of changes, the office which each performs is in this calculation fixed and invariable. Thus V_6 always prepares the numerators of the factors of any A; V_7 the denominators. V_8 always receives the (2n-3)th factor of {A}_{2n-1}, and V_9 the (2n-1)th. V_10 always decides which of two courses the succeeding processes are to follow, by feeling for the value of n 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.)

Supposing that it was desired not only to tabulate B_1, B_1, &c., but A_0, A_1, &c.; we have only then to appoint another series of Variables, V_41, V_42, &c., for receiving these latter results as they are successively produced upon V_11. 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: A_0, {A}_1{B}_1, {A}_3{B}_3, &c. We have then merely to multiply each B with each corresponding A, 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:— {d^{2}}/{d x^{2}}(z_{n+1} x^{2 n+2})=(2n+1)(2n+2) z^{n} x^{2n} for certain special suppositions respecting z, x and n.

The general integral itself is of the form, z_n=f(n).x+f_1(n)+f_2(n).x^{-1}+f_3(n).x^{-3}+... and it is worthy of remark, that the engine might (in a manner more or less similar to the preceding) calculate the value of this formula upon most other hypotheses for the functions in the integral, with as much, or (in many cases) with more, ease than it can formula (8.).

A. A. L.