Suppose the variables x, y … so connected that any one of the ratios x: y: z,… or, more generally, any determinate function of these ratios, depends on an equation of the µ th order. The variables x, y, z … are said to form a system of the µth order.

In the case of two variables x, y, supposing that these are connected by an equation U=O (U being a homogeneous function of the order µ) the variables form a system of the µth order; and, conversely, whenever the variables form a system of the µth order, they are connected by an equation of the above form.

In the case of a greater number of variables, the question is one of much greater difficulty. Thus with three variables x, y, z; if µ be resolvable into the factors then, supposing the variables to be connected by the equations U=0, V=0, U and V being homogeneous functions of the orders respectively, they will it is true form a system of the µth order, but the converse proposition does not hold: for instance, if µ is a prime number, the only mode of forming a system of the µth order would on the above principle be to assume that is to suppose the variables connected by an equation of the µth order and a linear equation; but this is far from being the most general method of obtaining such a system. In fact, systems not belonging to the class in question may be obtained by the introduction of subsidiary

This memoir was intended to appear at the same …