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LMNP I. Surfaces for which Da Dp Dv (a) (i) Ellipsoid, real, if —, — ' are all negative.

(ii) Hyperboloid of one sheet if two of the quantities are negative.

(iii) Hyperboloid of two sheets if one of the quantities is nega tive.

(iv) Ellipsoid, imaginary, if all the quantities are positive.

These are surfaces with centre. By a suit able transformation of the co-ordinate axes to the centre the general equation can be thrown into the form Xxt tiy2 vs2 d =-- 0, in which d = — and X, v are the roots of the discrirni nating cubic.

(B) D'='0, (i) Elliptic paraboloid if X and p have the same sign.

(ii) Hyperbolic paraboloid if and tz have different signs.

When D one of the roots X, v is zero, and it is here assumed that v O. By a suitable transformation of the origin of co ordinates to a point of the surface, the equa tion may be made to take the form Xx2 -1-py2 2Qs O.

The surfaces (a) and (ID are surfaces without vertices.

II. Surfaces for which d = O.

(y) D <0, (i) Cone, real, if A, II, v are not all of same sign.

(ii) Cone, imaginary, if v are of same sign.

The ruled surfaces among quadric surfaces fall into two categories: (1) the cone and cylinder, each containing one set of generating By taking the origin at the vertex of the cone the equation may be brought into the form Xxt i vist .= 0.

(5)D = 0 (i) Elliptic cylinder, if A and # are of same sign and O.

(ii) Hyperbolic cylinder, if A and IA are of different signs and d> O.

(iii) A pair of intersecting planes, real or imaginary, if d= O.

(iv) Parabolic cylinder; or two paral lel planes, real or imaginary (if all the subdeterminants of D are zero).

For (i), (ii), (iii), the general equation admits of being thrown into the form ± and for the parabolic cylinder into the form Si = O.

Since a cylinder may be regarded as a cone with infinitely distant vertex, and a pair of planes as a degenerate case of a cone, it follows that the surfaces (y) and (4) may be con sidered as cones, i.e., the surfaces with vertex or vertices.

lines infinite in number ; (2) the hyperboloid of one sheet and the hyperbolic paraboloid, each containing two sets of generating lines infinite in number. On the two last-named sur

faces no two lines of one set intersect, but each line of one set intersects all the lines of the other set.

5. Surfaces of the Third Order, or Cubic
considerable number of theorems about these surfaces are now known, though their properties have by no means been so ex haustively studied as in the case of quadric surfaces. There are two especially distinguish ing properties to be noted concerning them: first, on the general surface there are 27 right lines; second, there is related to the sur face a pentaedron whose edges and vertices lie on the Hessian of the surface. When there is no singular point on the cubic sur face its equation can be thrown into the form c2 W2' -I- cs We -I- kW +
0, where Wi, W2, . . . are linear in x, y, z and the equa tion Wri- WS+ W,+ W4 +
0 holds identi cally. The Hessian of the cubic is 1 1 1 1 1ci WI Cs Ws Cs Ws CS Ws Cs WS = 0, a surface of fourth order. It contains the 10 edges and the 10 vertices of the pentaedron formed by the five planes WP— 0. Wr=0, . . The vertices are double points (see 7) of the Hessian. For particulars as to these surfaces consult Salmon,

6. Surfaces of Fourth
Of these only special surfaces have been thoroughly dis cussed, among them the ruled surfaces and the Kummer surface. The Kummer surface contains 16 double points and 16 singular planes. Each of the planes is tangent to the surface along a conic and contains six of the double points, and through each double point pass six of the singular planes. Consult Salmon Fiedler, Vol. II, and Kummer,

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