There are three successive substitutions for each of these equations. The formulæ (2.), (3.), and (4.) are implicitly contained in (1.), which latter we may consider as being in fact the condensed expression of any of the former. It will be observed that every succeeding substitution must contain twice as many ’s as its predecessor. So that if a problem require substitutions, the successive series of numbers for the ’s in the whole of them will be 2, 4, 8, 16 ... . The substitutions in the preceding equations happen to be of little value towards illustrating the power and uses of the upper indices; for owing to the nature of these particular equations the indices are all unity throughout. We wish we had space to enter more fully into the relations which these indices would in many cases enable us to trace. M. Menabrea incloses the three centre columns of his table under the general title Variable-cards. The ’s however in reality all represent the actual Variable-columns of the engine, and not the cards that belong to them. Still the title is a very just one, since it is through the special action of certain Variable-cards (when combined with the more generalised agency of the Operation-cards) that every one of the particular relations he has indicated under that title is brought about. Suppose we wish to ascertain how often any one quantity, or combination of quantities, is brought into use during a calculation. [Pg 44] We easily ascertain this, from the inspection of any vertical column or columns of the diagram in which that quantity may appear. Thus, in the present case, we see that all the data, and all the intermediate results likewise, are used twice, excepting ), which is used three times. The order in which it is possible to perform the operations for the present example, enables us to effect all the eleven operations of which it consists, with only three Operation-cards; because the problem is of such a nature that it admits of each class of operations being performed in a group together; all the multiplications one after another, all the subtractions one after another, &c. The operations are .