1. IN what follows I malke use of the obvious principle, that, if two curves hatve a one to one correspondence, and if the points on a particular straight line of one correspond to points on a straight line of the other, the curves are of the same order and deficiency. The particular straight line is, in the cases I shall discuss, the line at infinity for both curves. 2. A curve may be determined as the locus of a point, which divides in a constant ratio a terminated straight line variable in length and position; and a family of curves related to one another is obtained by changing thie ratio. Each such family depends, therefore, on one parameter. An additional parameter is introduced by the transformation of which I propose to treat. There are commonly particular values of the ratio, which give curves of a lower order than that of the general locus, especially by the reduplication of the locus or a curve-factor of the locus. For example, if straight lines tllrotughi a point meet a circle, the locus of the middle points of the iiitercepts by the circle is another circle through the fixed point, and the centre of the direction. This is plainly a special locus, and if the chords are divided by the generating point in the ratio k: 1, the locus is found to be the inverse of a conic, i. e. of the fourth order. We may look for a similar reduction of the order whenever the ratio is one of equality, and tlle extremities.of the linear segments move on one and the same curve. It is to be understood that when I hereafter speak of a curve of tlhe class described, which for' brevity I shall call a ratio-curve, I mean one general as to order; in fact' I suppose the ratio of division to bd denoted by general literal symbols, to which such values may be ascribed as will not give rise to a special reduction of order.