Tangential Equations Research Articles

Foci in Complex Geometry

The object of this paper is to indicate the part played by the concept 'focus' in the geometry of the Gauss plane. The relation between focal properties and complex geometry is a reciprocal one. On the one hand, the method of complex representation may be utilised to elucidate the properties of real foci of curves; this is what is done in para. 1 of the paper, where a general theorem is stated, which gives immediately the position of the real foci of a curve given by a real areal tangential equation.

On the stationary eccentricity of a system of four-tangent conics

Let the convex quadrilateral formed by the four given tangents be ABA′B′, and O the intersection of the diagonals. Let OA and OB be taken as axes of x and y. Denote OA, OA′, OB and OB′ by a, a′, b and b′, a and b being positive, and a′ and b′ negative. The tangential equation of the system is thenwhere k is a variable parameter; for the equation is satisfied when the straight line lx + my + 1 = 0 passes through any two adjacent angular points of the quadrilateral.

Curves without Double Points

MR. BASSET'S objection to the term “non-singular” (see NATURE, Oct. 11, p. 572), arises from a misunderstanding. The ordinary use ot the term by English-speaking mathematicians is natural and legitimate; it is applied to curves without double points when the curve in question is defined by a relation among the coordinates of its points. In the case of a curve defined in another manner, for instance by a tangential equation, “non-singular” could not possibly be used in the sense. In fact, the phrase which Mr. Basset denounces as “exceedingly infelicitous and misleading” is one which, standing by itself as Mr. Basset quotes it, strikes a geometrician as unfamiliar; “non-singular cubic curve,” “non-singular curve of the nth order,” are familiar to him, and are un-objectionable.

XI. On a new form of tangential equation

Art. 1. The tangential equation of a curve is, as is well known, a relation among the coefficients in the equation of a variable line, which being fulfilled, the line must be a tangent to the curve. Let O be the origin, OX, OY the axes; and let a variable line MN in any of its positions make an intercept ⋎ on OX and an angle ϕ with it; then the equation of the line is x + y cot ϕ - ⋎ = 0, and ⋎ and ϕ , the quantities which determine the position of the line, may be called its coordinates. From this it follows that any relation between ⋎ and ϕ , such as ⋎ = f ( ϕ , .............. (1) will be the tangential equation of a curve which is the envelope of the line.

III. On a new form of tangential equation

If a variable line make an intercept v on the axis of x , and an angle ϕ with it on the negative side, the equation of this line will be x+y cot ϕ-v =0. The quantities v and ϕ will determine the position of the line, and may therefore be called its coordinates; hence any relation between v and ϕ such as v = f(ϕ) , will be the tangential equation of a curve which is the envelope of the line.

XII. On the bicircular quartic.—Addition to Professor Casey’s memoir "On a new form of tangential equation."

Professor Casey communicated to me the MS. of the foregoing Memoir, and he has permitted me to make to it the present Addition, containing further developments on the theory of the bicircular quartic.