Thus the terms of the same degree, however different may be their composition, varying in the same manner, and the terms of different degrees varying in an unequal proportion, whatever similarity there may be in their composition, it will be necessary, to prevent the equation from being disturbed, that all the terms which it contains should be of the same degree. It is in this that properly consists the ordinary theorem of Homogeneity, and it is from this circumstance that the general law has derived its name, which, however, ceases to be exactly proper for all other functions.
In order to treat this subject in its whole extent, it is important to observe an essential condition, to which attention must be paid in applying this property when the phenomenon expressed by the equation presents magnitudes of different natures. Thus it may happen that the respective units are completely independent of each other, and then the theorem of Homogeneity will hold good, either with reference to all the corresponding classes of quantities, or with regard to only a single one or more of them. But it will happen on other occasions that the different units will have fixed relations to one another, determined by the nature of the question; then it will be necessary to pay attention to this subordination of the units in verifying the homogeneity, which will not exist any longer in a purely algebraic sense, and the precise form of which will vary according to the nature of the phenomena. Thus, for example, to fix our ideas, when, in the analytical expression of geometrical phenomena, we are considering at once lines, areas, and volumes, it will be necessary to observe that the three corresponding units are necessarily so connected with each other that, according to the subordination generally established in that respect, when the first becomes m times greater, the second becomes m2 times, and the third m3 times. It is with such a modification that homogeneity will exist in the equations, in which, if they are algebraic, we will have to estimate the degree of each term by doubling the exponents of the factors which correspond to areas, and tripling those of the factors relating to volumes.
Such are the principal general considerations relating to the Calculus of Direct Functions. We have now to pass to the philosophical examination of the Calculus of Indirect Functions, the much superior importance and extent of which claim a fuller development.