Arthur Cayley and Edwin Bailey Elliott

“ Forms of Real Curves.—We have in much that precedes disregarded, or at least been indifferent to, reality; it is only thus that the conception of a curve of the m-th order, as one which is met by every right line in m points, is arrived at; and the curve itself, and the line which cuts it, although both are tacitly assumed to be real, may perfectly well be imaginary. For real figures we have the general theorem that imaginary intersections, &c., present themselves in conjugate pairs ”

“ Analogous to the order and class of a plane curve we have the order, rank and class of the system (assumed to be a geometrical one) , viz. if an arbitrary plane contains m points, an arbitrary line meets r lines, and an arbitrary point lies in n planes, of the system, then m, r, n are the order, rank and class respectively. The system has singularities, and there exist between m, r, n and the numbers of the several singularities equations analogous to Plücker’s equations for a plane curve. ”

“ First, if the three intersections by the line infinity are all distinct, we have the hyperbolas; if the points are real, the redundant hyperbolas, with three hyperbolic branches; but if only one of them is real, the defective hyperbolas, with one hyperbolic branch. Secondly, if two of the intersections coincide, say if the line infinity meets the curve in a onefold point and a twofold point, both of them real, then there is always one asymptote: the line infinity may at the twofold point touch the curve, and we have the parabolic hyperbolas ”

“ The leg of an infinite branch may have at the extremity a tangent; this is an asymptote of the curve, and the leg is then hyperbolic; or the leg may tend to a fixed direction, but so that the tangent goes further and further off to infinity, and the leg is then parabolic; a branch may thus be hyperbolic or parabolic as to its two legs; or it may be hyperbolic as to one leg and parabolic as to the other. The epithets hyperbolic and parabolic are of course derived from the conic hyperbola and parabola respectively. ”

“ In a machine of any kind, each point describes a curve; a simple but important instance is the “three-bar curve,” or locus of a point in or rigidly connected with a bar pivoted on to two other bars which rotate about fixed centres respectively. ”

“ The oval may unite itself with the infinite branch, or it may dwindle into a point, and we have the crunodal and the acnodal forms respectively; or if simultaneously the oval dwindles into a point and unites itself to the infinite branch, we have the cuspidal form. (See Parabola.) Drawing a line to cut any one of these curves and projecting the line to infinity, it would not be difficult to show how the line should be drawn in order to obtain a curve of any given species. ”

“ The branch, whether re-entrant or infinite, may have a cusp or cusps, or it may cut itself or another branch, thus having or giving rise to crunodes or double points with distinct real tangents; an acnode, or double point with imaginary tangents, is a branch by itself,—it may be considered as an indefinitely small re-entrant branch. a branch may have inflections and double tangents, or there may be double tangents which touch two distinct branches; there are also double tangents with imaginary points of contact, which are thus lines having no visible connexion with the curve. ”

“ An even circuit not cutting itself divides the plane into two parts, the one called the internal part, incapable of containing any odd circuit, the other called the external part, capable of containing an odd circuit.
We may now state in a more convenient form the fundamental distinction of the kinds of cubic curve. A non-singular cubic is simplex, consisting of one odd circuit, or it is complex, consisting of one odd circuit and one even circuit. It may be added that there are on the odd circuit three inflections, but on the even circuit no inflection ”

“ The expression for the number of inflections 3m (m − 2) for a curve of the order m was obtained analytically by Plücker, but the theory was first given in a complete form by Hesse in the two papers “Über die Elimination, u.s.w.,” and “Über die Wendepuncte der Curven dritter Ordnung” (Crelle, t. xxviii., 1844) ; in the latter of these the points of inflection are obtained as the intersections of the curve u = 0 with the Hessian, or curve Δ = 0, where Δ is the determinant formed with the second derived functions of u. We have in the Hessian the first instance of a covariant of a ternary form. ”

“ A curve of a given order can in general be rationally transformed into a curve of a lower order; thus a curve of any order for which D = 0, that is, a unicursal curve, can be transformed into a line; a curve of any order having the deficiency 1 or 2 can be rationally transformed into a curve of the order D + 2, deficiency D ”

“ We have herein a better principle of classification; considering cubic curves, in the first instance, according to singularities, the curves are non-singular, nodal (viz. crunodal or acnodal) , or cuspidal; and we see further that there are two kinds of non-singular curves, the complex and the simplex. There is thus a complete division into the five kinds, the complex, simplex, crunodal, acnodal and cuspidal. Each singular kind presents itself as a limit separating two kinds of inferior singularity ”

“ A non-singular quartic has only even circuits; it has at most four circuits external to each other, or two circuits one internal to the other, and in this last case the internal circuit has no double tangents or inflections. A very remarkable theorem is established as to the double tangents of such a quartic: distinguishing as a double tangent of the first kind a real double tangent which either twice touches the same circuit, or else touches the curve in two imaginary points, the number of the double tangents of the first kind of a non-singular quartic is = 4 ”

“ Hence, in the case of a curve which has δ nodes and κ cusps, the apparent diminution 2 (m2 − m − 6) (2δ + 3κ) is too great, and it has in fact to be diminished by 2{2δ (δ − 1) + 6δκ + 9/2κ (κ − 1) }, or the half thereof is 4 for each pair of nodes, 6 for each combination of a node and cusp, and 9 for each pair of cusps. We have thus finally an expression for 2τ, = m (m − 2) (m2 − 9) − &c.; or dividing the whole by 2, we have the expression for τ given by the third of Plücker’s equations. ”

“ Considering the variable curve corresponding to a given value of the parameter, or say simply the variable curve, the consecutive curve has then also δ and κ nodes and cusps, consecutive to those of the variable curve; and it is easy to see that among the intersections of the two curves we have the nodes each counting twice, and the cusps each counting three times; the number of the remaining intersections is = m2 − 2δ − 3κ. ”

“ Connecting with each theory the author’s name, the theories in question are G. F. B. Riemann, the rational transformation of a plane curve; Luigi Cremona, the rational transformation of a plane; and Chasles, correspondence of points on the same curve, and united points. The theory first referred to, with the resulting notion of “Geschlecht,” or deficiency, is more than the other two an essential part of the theory of curves, but they will all be considered.
Riemann’s results are contained in the memoirs on “Abelian Integrals,” &c. ”

“ Duality. Trilinear and Tangential Co-ordinates.—The Géométrie descriptive, by Gaspard Monge, was written in the year 1794 or 1795 (7th edition, Paris, 1847) , and in it we have stated, in plano with regard to the circle, and in three dimensions with regard to a surface of the second order, the fundamental theorem of reciprocal polars, viz. “Given a surface of the second order and a circumscribed conic surface which touches it ... then if the conic surface moves so that its summit is always in the same plane, the plane of the curve of contact passes always through the same point.” ”

“ The principle in its original form as applying to a right line was used throughout by Chasles in the investigations on the number of the conics which satisfy given conditions, and on the number of solutions of very many other geometrical problems.
There is one application of the theory of the (α, α′) correspondence between two planes which it is proper to notice. ”

“ The elementary problems in regard to cubics are solved very completely by S. Maillard in his Thèse, Recherche des caractéristiques des systèmes élémentaires des courbes planes du troisième ordre (Paris, 1871) . ”

“ R. F. A. Clebsch and C. L. F. Lindemann, Vorlesungen über Geometrie, Band i. and i2 (Leipzig, 1875–1876) ; H. Schroeter, Die Theorie der ebenen Kurven dritter Ordnung (Leipzig, 1888) ; H. Andoyer, Leçons sur la théorie des formes et la géométrie analytique supérieure (Paris, 1900) ; Wieleitner, Theorie der ebenen algebraischen Kurven höherer Ordnung (Leipzig, 1905) . (A. Ca.; E. B. El.)
↑ In solid geometry infinity is a plane—its intersection with any given plane being the right line which is the infinity of this given plane. ”