We thus see that when , nine Operation-cards are used; that when , fourteen Operation-cards are used; and that when 2" data-tex="n \gt 2">, twenty-five Operation-cards are used; but that no more are needed, however great may be; and not only this, but that these same twenty-five cards suffice for the successive computation of all the Numbers from , to , 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 will require twenty-seven Variable-cards; forty-two such cards; seventy-five; and for every succeeding after , 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 ). 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 [Pg 64] 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 after . 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.