it being required that the result shall be developed, like the dividend and the divisor, in successive powers of . A little consideration of (1.), and of the steps through which algebraical division is effected, will show that (if the denominator be supposed to consist of terms) the first partial quotient will be completed by the following operations:— that the second partial quotient will be completed by an exactly similar set of operations, which acts on the remainder obtained by the first set, instead of on the original dividend. The whole of the processes therefore that have been gone through, by the time the second partial quotient has been obtained, will be,— which is a cycle that includes a cycle, or a cycle of the second order. The operations for the complete division, supposing we propose to obtain terms of the series constituting the quotient, will be,— It is of course to be remembered that the process of algebraical division in reality continues ad infinitum, except in the few exceptional cases which admit of an exact quotient being obtained. The number in the formula (4.), is always that of the number of terms we propose to ourselves to obtain; and the th partial quotient is the coefficient of the ()th power of . There are some cases which entail cycles of cycles of cycles, to an indefinite extent. Such cases are usually very complicated, and they are of extreme interest when considered with reference to the engine. The algebraical development in a series, of the th function of any given function, is of this nature. Let it be proposed to obtain the th function of We should premise that we suppose the reader to understand what is meant by an th function. We suppose him likewise to comprehend distinctly the difference between developing an th function algebraically, and merely calculating an th function arithmetically. If he does not, the following will be by no means very intelligible; but we have not space to give any preliminary explanations. To proceed: the law, according to which the successive functions of (5.) are to be developed, must of course first be fixed on. This law may be of very various kinds. We may propose to obtain our results in successive powers of , in which case the general form would be [Pg 52] or in successive powers of itself, the index of the function we are ultimately to obtain, in which case the general form would be and would only enter in the coefficients. Again, other functions of or of instead of powers, might be selected. It might be in addition proposed, that the coefficients themselves should be arranged according to given functions of a certain quantity. Another mode would be to make equations arbitrarily amongst the coefficients only, in which case the several functions, according to either of which it might be possible to develop the th function of (5.), would have to be determined from the combined consideration of these equations and of (5.) itself.