7. Ordinary and Singular In gen eral the n points of intersection of a line and surface are distinct, but if a point P appears as k coincident points among the n intersec tions for every straight line through it, it is called a k multiple point of the surface. When k =2 it is a double point, when k— 3 a triple point. Multiple points are singular points. Through a double point on a surface an in finity of lines can be drawn each of vThich will pass through a third point on the surface in finitely near the double point. The locus of these lines is a cone of second order. There are three cases: (1) the cone is a proper cone and the double point is called a conical point; (2) the cone degenerates into two intersecting planes and the point is a biplanar point; (3) the cone degenerates into two coincident planes and the point is a uniplanar point or pinch point. There may be a curve locus of double points on a surface: for example, the curve of intersection of a surface by itself is such a locus, as is also the curve of contact of two sheets of the same surface with each other. For a detailed study of the character of a sur face in the neighborhood of a double point consult Rohn, Mathematische Annalen, (1883). As the study of a surface proceeds in general from the equation of the surface, it is to the analysis we must look for a definitive criterion distinguishing ordinary and singular points. A point xi, zi of a surface y, z) =0 is be ordinary, if the F function is developable in an entire series in the neighbor hood of xi,and if the three first deriva 8F oF aF tives -, as — do not simultaneously vanish at the point. All other points are singular points.
8. General Considerations. Curvilinear Co ordinates.— Modern progress in the theory of surfaces begins with the appearance in 1827 of Gauss' paper, (Disquisitiones generates circa superficies curvas) (translated into German in Ostwald's (Klassiker der exakten Wissenschaf ten' ; into English by Morehead and Hilte beitel, Princeton). Two things in this classical production have profoundly affected subse quent developments in the theory. The first was the systematic employment of curvilinear co-ordinates, and therewith a demonstration of the great advantages which could be derived from their use; the second was the conception of a surface as a two-way extension, not rigid but flexible, which could be made to assume new shapes by bending without stretching. All surfaces derived from a given surface by bend ing are said to be applicable, or developable, upon each other. It is clear that the geometry of figures on such surfaces is the same. The analytical criteria, whether two given surfaces are applicable upon each other, constitute one of the interesting chapters in the general theory. In expressing the Cartesian co-ordinates x, y, z as functions of two variables u, v called parameters: (A) x 0(u, v), y x(u, v), =- Ifr(u, v), a new form of representation of surfaces is established. The elimination of u and v would
obviously lead to one equation, F(x, y, If u be given a definite value no, and v be al lowed to vary, a curve will be generated lying on the surface (see CURVES OF DOUBLE CURVA TURE). The curve is called the uo curve, that is, it is named by the special value of the para meter that is constant at all its points. Assign ing a second value to u, say u,, and allowing v again to vary, there would be formed the m curve of the surface. In this way there could be formed an api of curves on the surface con stituting the family of u curves. Similarly, there is a family of v curves each characterised by a definite value of v while u is variable. Each point of the surface is the intersection of a u curve and a v curve, and the curves are called its curvilinear One may thus speak of the point of the surface, or in general of the point (u, v), in place of referring to it by its Cartesian co-ordinates. Both co-ordinate systems are put in evidence by writing the point in the form (x, y, z; u, v). A restriction upon the values u and v may take, such as an eauation f(u, v) =0, defines a curve on the surface.
9. Tangent Plane. Principal Normal Sec tions.— If to all the curves on a surface passing through an ordinary point P (x, y, z; u, v), tangents be drawn at P, the line will lie in a plane called the tangent plane at P. Its equation is lay az ay as ( _ lax az- as ax Fzka7,1v a74) 5 — 4 + — TO (' — kau a, au au ay)(c — 0 = 0. + (-- -- au av av au where f, 07, C are the current co-ordinates of the points of the plane.
The line perpendicular to the tangent plane at P is the normal of the surface at P. Every plane through the normal is a normal plane, and the sections made by them with the sur face are normal sections.
10. The Fundamental Quadratic Forms and the Fundamental Magnitudes of the First and Second Order. The Fundamental Equations. —The entire theory of a surface is implicitly involved in two fundamental quadratic differ ential forms, and in three fundamental differ ential equations. The first form is the ex pression for the square of the arc length ds on the surface between two infinitely near points P.(x, y, z; u v) and P, (x + dx, y + dy, z+ dz; u du, v+ dv). This is found to be (a) + Fdu dv + Gd?, where E_ ax\' lay\ _+ tas \ 2 al‘i -1- 44 kaUi ' , aX Oy as as __. __..• au av au av au av The second quadratic differential form is the expression for twice the distance, d, from the point P, to the tangent plane at P: (fi) Ddu'+ 2D du dv + D"dv', where a'y a's Mx a's au= au= au= ao az2 al,2 ax ay as ax ay az au du du au au au dx ay az ax ay az D av avav ay av D" —,