No notice has yet been taken of the upper indices which are added to the left of each in the diagram; an addition which we have also taken the liberty of making to the ’s in M, Menabrea’s tables of pages 16, 19, since it does not alter anything therein represented by him, but merely adds something to the previous indications of those tables. The lower indices are obviously indices of locality only, and are wholly independent of the operations performed or of the results obtained, their value continuing unchanged during the performance of calculations. The upper indices, however, are of a different nature. Their office is to indicate any alteration in the value which a Variable represents; and they are of course liable to changes during the processes of a calculation. Whenever a Variable has only zeros upon it, it is called ; the moment a value appears on it (whether that value be placed there arbitrarily, or appears in the natural course of a calculation), it becomes . If this value gives place to another value, the Variable becomes , and so forth. Whenever a value again gives place to zero, the Variable again becomes , even if it have been the moment before. If a value then again be substituted, the Variable becomes (as it would have done if it had not passed through the intermediate ); &c. &c. Just before any calculation is commenced, and after the data have been given, and everything adjusted and prepared for setting the mechanism in action, the upper indices of [Pg 43] the Variables for data are all unity, and those for the Working and Result-variables are all zero. In this state the diagram represents them[24]. There are several advantages in having a set of indices of this nature; but these advantages are perhaps hardly of a kind to be immediately perceived, unless by a mind somewhat accustomed to trace the successive steps by means of which the engine accomplishes its purposes. We have only space to mention in a general way, that the whole notation of the tables is made more consistent by these indices, for they are able to mark a difference in certain cases, where there would otherwise be an apparent identity confusing in its tendency. In such a case as there is more clearness and more consistency with the usual laws of algebraical notation, in being able to write . It is also obvious that the indices furnish a powerful means of tracing back the derivation of any result; and of registering various circumstances concerning that series of successive substitutions, of which every result is in fact merely the final consequence; circumstances that may in certain cases involve relations which it is important to observe, either for purely analytical reasons, or for practically adapting the workings of the engine to their occurrence. The series of substitutions which lead to the equations of the diagram are as follow:—