The question of the analysis of Higher Singularities is one whose importance has been recognized from the time of Cramer. Two of the principal methods of dealing with it - by expansions, and by successive quadric transformations - are used in his "Analyse des Lignes Courbes"; the explanation of the principles involved, and the development of the theory, belong to the present day. The memoirs to which reference is made in this paper are those by Cayley, "On the Higher Singularities of a Plane Curve," 1866, and H J Smith, "On the Higher Singularities of Plane Curves," 1873-6, and the series of papers by Brill and Nöther in the 'Mathematische Annalen', etc.

In Vol. XIV of this Journal I gave an account of a geometrical method of analysing Higher Singularities, by means of which there may be found for any singularity a penultimate form involving a series of nodes with a certain number of evanescent loops. It was there stated that the method is directly applicable, in general, to the determination only of the point components of the singularity, though in certain cases it determines the inflexions. I propose now to remove this restriction, showing that the process enables us, in every case, to enumerate the double lines (double tangents and inflexional tangents) involved in the singularity.

In the following pages I attempt to show, as a matter of purely pedagogic interest, how simply and naturally Cayley's theory of the Absolute follows from a small number of very elementary geometrical conceptions, without any appeal to analytical geometry. Where assumptions are made, the fact is frankly stated; the few points where more advanced mathematical reasoning is needed for the actual proof are clearly indicated ; my contention is not that every step in the rigorous proof can be presented under the guise of elementary mathematics, but that it is quite possible to develop the theory so as to be intelligible and interesting to average students at a much earlier stage than is customary.

The theory of the intersections of curves has probably led its investigators into more errors than any other modern theory. Even the history of the central question, the so-called Cramer paradox, is usually given incorrectly, with the omission of all reference to Maclaurin. This is the more surprising, inasmuch as Cramer himself ascribes to Maclaurin the theorem that the $n^{2}$ intersections of two curves of order $n$ impose only a certain number of conditions, and gives the exact reference. Plücker was familiar with the passage in Cramer; nevertheless he overlooks Maclaurin, though he was sufficiently interested historically in the Cramer paradox to give also the reference to Euler.