The power of repeating the cards, alluded to by M. Menabrea in page 15, and more fully explained in Note C., reduces to an immense extent the number of cards required. It is obvious that this mechanical improvement is especially applicable wherever cycles occur in the mathematical operations, and that, in preparing data for calculations by the engine, it is desirable to arrange the order and combination of the processes with a view to obtain them as much as possible symmetrically and in cycles, in order that the mechanical advantages of the backing system may be applied to the utmost. It is here interesting to observe the manner in which the value of an analytical resource is met and enhanced by an ingenious mechanical contrivance. We see in it an instance of one of those mutual adjustments between the purely mathematical and the mechanical departments, mentioned in Note A. as being a main and essential condition of success in the invention of a calculating engine. The nature of the resources afforded by such adjustments would be of two principal kinds. In some cases, a difficulty (perhaps in itself insurmountable) in the one department, would be overcome by facilities in the other; and sometimes (as in the present case) a strong point in the one, would be rendered still stronger and more available, by combination with a corresponding strong point in the other. As a mere example of the degree to which the combined systems of cycles and of backing can diminish the number of cards requisite, we shall choose a case which places it in strong evidence, and which has likewise the advantage of being a perfectly different kind of problem from those that are mentioned in any of the other Notes. Suppose it be required to eliminate nine variables from ten simple equations of the form— We should explain, before proceeding, that it is not our object to consider this problem with reference to the actual arrangement of the [Pg 55] data on the Variables of the engine, but simply as an abstract question of the nature and number of the operations required to be performed during its complete solution. The first step would be the elimination of the first unknown quantity between the two first equations. This would be obtained by the form— for which the operations 10 () would be needed. The second step would be the elimination of , between the second and third equations, for which the operations would be precisely the same. We should then have had altogether the following operations:— Continuing in the same manner, the total number of operations for the complete elimination of between all the successive pairs of equations, would be— We should then be left with nine simple equations of nine variables from which to eliminate the next variable ; for which the total of the processes would be— We should then be left with eight simple equations of eight variables from which to eliminate , for which the processes would be—