_Another equivalent distinction._ Still farther, when we examine more profoundly this distinction of different orders of differential equations, we find that it can be always made to come under a final general distinction, relative to differential equations, which remains to be noticed. Differential equations with one or more independent variables may contain simply a single function, or (in a case evidently more complicated and more implicit, which corresponds to the differentiation of simultaneous implicit functions) we may have to determine at the same time several functions from the differential equations in which they are found united, together with their different derivatives. It is clear that such a state of the question necessarily presents a new special difficulty, that of separating the different functions desired, by forming for each, from the proposed differential equations, an isolated differential equation which does not contain the other functions or their derivatives. This preliminary labour, which is analogous to the elimination of algebra, is evidently indispensable before attempting any direct integration, since we cannot undertake generally (except by special artifices which are very rarely applicable) to determine directly several distinct functions at once. Now it is easy to establish the exact and necessary coincidence of this new distinction with the preceding one respecting the order of differential equations. We know, in fact, that the general method for isolating functions in simultaneous differential equations consists essentially in forming differential equations, separately in relation to each function, and of an order equal to the sum of all those of the different proposed equations. This transformation can always be effected. On the other hand, every differential equation of any order in relation to a single function might evidently always be reduced to the first order, by introducing a suitable number of auxiliary differential equations, containing at the same time the different anterior derivatives regarded as new functions to be determined. This method has, indeed, sometimes been actually employed with success, though it is not the natural one.