We should observe, that if the variables , , , &c. are furnished, they would be placed directly upon , , &c., like any other data. If not, a separate computation might be entered upon in a separate part of the engine, in order to calculate them, and place them on , &c. We have now explained how the engine might compute (1.) in the most direct manner, supposing we knew nothing about the general term of the resulting series. But the engine would in reality set to work very differently, whenever (as in this case) we do know the law for the general term. The two first terms of (1.) are and the general term for all after these is which is the coefficient of the ( term. The engine would calculate the two first terms by means of a separate set of suitable Operation-cards, and would then need another set for the third term; which last set of Operation-cards would calculate all the succeeding terms ad infinitum; merely requiring certain new Variable-cards for each term to direct the operations to act on the proper columns. The following would be the successive sets of operations for computing the coefficients of terms— Or we might represent them as follows, according to the numerical order of the operations:— The brackets, it should be understood, point out the relation in which the operations may be grouped, while the comma marks succession. The symbol + might be used for this latter purpose, but this would be liable to produce confusion, as + is also necessarily used to represent one class of the actual operations which are the subject of that succession. In accordance with this meaning attached to the comma, care must be taken when any one group of operations recurs more than once, as is represented above by (11 ... 15), not to insert a comma after the number or letter prefixed to that group. ,(11 ... 15) would stand for an operation followed by the group of operations (11 ... 15); instead of denoting the number of groups which are to follow each other. Wherever a general term exists, there will be a recurring group of operations, as in the above example. Both for brevity and for distinctness, a recurring group is called a cycle. A cycle of operations, then, must be understood to signify any set of operations which is repeated more than once. It is equally a cycle, whether it be repeated twice only, or an indefinite number of times; for it is the fact of a repetition occurring at all that constitutes it such. In many cases of analysis there is a recurring group of one or more [Pg 51] cycles; that is, a cycle of a cycle, or a cycle of cycles. For instance: suppose we wish to divide a series by a series,