2. By means of certain artifices and arrangements (upon which we cannot enter within the restricted space which such a publication as the present may admit of), there is no limit either to the magnitude of the numbers used, or to the number of quantities (either variables or constants) that may be employed. 3. It can combine these numbers and these quantities either algebraically or arithmetically, in relations unlimited as to variety, extent, or complexity. 4. It uses algebraic signs according to their proper laws, and developes the logical consequences of these laws. 5. It can arbitrarily substitute any formula for any other; effacing the first from the columns on which it is represented, and making the second appear in its stead. 6. It can provide for singular values. Its power of doing this is referred to in M. Menabrea’s memoir, page 20, where he mentions the passage of values through zero and infinity. The practicability of causing it arbitrarily to change its processes at any moment, on the occurrence of any specified contingency (of which its substitution of for () explained in Note E., is in some degree an illustration), at once secures this point. The subject of integration and of differentiation demands some notice. The engine can effect these processes in either of two ways:— First. We may order it, by means of the Operation and of the Variable-cards, to go through the various steps by which the required limit can be worked out for whatever function is under consideration. Secondly. It may (if we know the form of the limit for the function in question) effect the integration or differentiation by direct[28] substitution. We remarked in Note B., that any set of columns [Pg 58] on which numbers are inscribed, represents merely a general function of the several quantities, until the special function have been impressed by means of the Operation and Variable-cards. Consequently, if instead of requiring the value of the function, we require that of its integral, or of its differential coefficient, we have merely to order whatever particular combination of the ingredient quantities may constitute that integral or that coefficient. In , for instance, instead of the quantities being ordered to appear on in the combination , they would be ordered to appear in that of They would then stand thus:— Similarly, we might have , the integral of . An interesting example for following out the processes of the engine would be such a form as or any other cases of integration by successive reductions, where an integral which contains an operation repeated times can be made to depend upon another which contains the same or times, and so on until by continued reduction we arrive at a certain ultimate form, whose value has then to be determined.