Quarterly Journal of Pure and Applied Mathematics, Vol. 28 (1896), 311-374

Introduction

The Theory of Singular Solutions is now generally understood to relate to certain solutions of a rational, integral, algebraic equation of the *n*^{th} degree in *p*, (*dy/dx*), whose coefficients are rational, integral, algebraic functions of *x, y*. The primitive of this equation is known to be a rational, integral, algebraic equation of the *n*^{th} degree in an arbitrary constant, Ω, having coefficients which are functions, not necessarily algebraic, of *x, y*. A solution of the *p*-equation which cannot be deduced from the primitive by giving Ω any special value is called a "singular solution"; a solution which can be so deduced is called a "particular integral." (We shall denote this by the letters P.I.).

The singular solution is known to be the envelope of the family of curves represented by the complete primitive; hence the geometrical statement of the above is: A rational, integral, algebraic equation of the *n*^{th} degree in *p*, whose coefficients are rational, integral, algebraic functions of *x, y*, represents a family of curves, not necessarily algebraic, depending on the arbitrary parameter Ω, and such that through every point of the plane there pass *n* curves of the family. The equation of an envelope of this family of curves satisfies the *p*-equation and is called a "singular solution"; any particular curve of the family is a "particular integral."

If the coordinates of any particular point be substituted for *x, y* in the Ω- and *p*-equations these equations give respectively, the *n* values of Ω which determine the *n* curves through the point, and the *n* values of *p* which determine the directions of the tangents to the *n* curves at the point. If the particular point be on the envelope, two of the curves through the point are *consecutive*, and the directions of the tangents to these two curves are *consecutive*, hence each of the equations has a pair of equal roots, that is, the envelope is the locus of points at which the Ω-equation has a pair of equal (consecutive) roots in Ω, and the *p*-equation has a pair of equal (consecutive) roots in *p*.

But regarding the equations as binary quantics in Ω/1 and *p*/1 respectively, the discriminant of the Ω-equation is the locus of points at which two values of Ω are equal, and the discriminant of the *p*-equation is the locus of points at which two values of *p* are equal; therefore we get, as a general result which will be modified later, *a common factor of the* Ω- *and p-discriminants is a singular solution of the p-equation and gives the envelope of the family of curves represented by it.*