SURFACES OF THE SECOND DEGREE. This name is given to all those surfaces of which the equation is of the second degree, or can be made a case of to which form any equation of the second degree between three variables may be reduced. These surfaces hold the same place among surfaces which is held by curves of the second degree, or conic sections, among curves ; and every section made by a plane with any surface of the second degree must be a curve of the second degree. The following article is intended entirely for reference, as tho books which treat on ' the subject hardly ever give the complete tests for the separation of the different cases from each other.
1. The preceding equation may be wholly impossible, or incapable of being satisfied by any values of x, y, and z. This happens when the left hand side can be resolved into the sum of any number of squares which cannot vanish simultaneously.
2. It may represent only one single point. In this case the left-hand side can be resolved into the sum of three squares, which vanish simultaneously for one set of values of x, y, and z.
3. The equation may belong to a single straight line. In this case the left-hand aide can be resolved into the sum of two squares.
4. The two last cases have a particular case which is algebraically very distinguishable from the rest, though it can only be geometrically represented by saying that the point or line is at an infinite distance from the origin.
5. The equation may belong to a single plane. In this case the left-hand side is a perfect square.
6. Or to a pair of planes, either parallel or intersecting. The left band Ride can then be resolved into two different factors of the first degree.
In the preceding cams there is no other surface than can be repre sented by one or several equations of the first degree. We now come to the cases in which new surfaces, not plane, are generated. But we may first observe that the left-hand side of the equation has a property much resembling a celebrated one of integer numbers. If it be the sum of any number of squares exceeding four, it may be reduced to the sum of four squares at most.
7. The equation may belong to a cone, having for its base any one of the conic sections. But in every case the same cone may be described by a circle only : that is, every cone of the second order is a circular cone, right or oblique. In this case, the first side of the equation takes the form e- + or e - Q, and it being expressions of the first e, of the form ar +By+ cz + E.
8. The equation may belong to a cylinder having for its base any conic section. But the elliptic, parabolic, and hyperbolic cylinders are perfectly distinct. In this case the first side of the equation can be reduced to the form r and Q being expressions of the first degree, and 74 and n constants.
D. The equation may belong to an ellipsoid, a single hyperboloid, double hyperboloid, an elliptic paraboloid, or an hyperbolic paraboloid. These five are the distinct surfaces of the second degree, answering to the three distinct curves of the second degree—namely, the ellipsoid to the ellipse, the two hyperboloids to the hyperbola, and the two paraboloids te the parabola. They will presently be further described ; in the mean time the forms to which the left-hand side of the equation may he reduced in these several cases are— Ell I peoid + + — Single If yperboloid + - Ri — Double ilyperbeloid - Elliptic Paraboloid el+ + Hyperbolic Paraboloid +Inn.
The conditions under which 1the several cases are produced are exhibited in the following table. Let v,= ahe+2e'L'c' — aa"— Lb" —eel v (he — + fab— +2 (b'c'—aa') L 2 (c'a'— c"a" + 2 (a'11—ec') a"b"v, w = — — + f When 0, is a perfect square: if also= 0, the three expression 2cia'b"+ al," ac"— 2b'e"a" +ea" + be" ac— t/2 • ea — be—al are all equal. Let either of them, with its sign changed, and increase( by f, be called w'. Again, when any three of the six quantities be— am, ab — c': — art', e'a' — bh', a'b' — cc' vanish, the other three also vanish. Let these vanish, and also let a'
1." be in the proportion of a, c', 1,', or of c', b, a', or of b', a', e. Whet this happens, the three following a" 11" a 1.) c are equal : let either, with its sign changed, and increased by f, b called w". The table is then as follows, in which p means either the signs + or —, and n means the other; and a supposition put h For example, it is the condition of an ellipsoid that w and v, should be finite with different signs, that v, should be positive, and v, of the mile sign as v,: it is the condition of intersecting planes that w should have the form 0+0, or that v, and v, should both vanish : that w' mbould also vanish; and that v, should be negative. It is the condition a single hyperboloid, if v, be positive, that w and v, should both ilffer in sign from v, ; but if be negative, it is enough that w and v, should have the same sign. All that precedes is equally true whether the co-ordinates be oblique or rectangular ; but the following is only true for rectangular co-ordinates : if the surface he a surface of revolution, it is necessary that b'c'—aa' ea' —Lb —cc' a' = The forms of the ellipsoid and of the two hyperboloids may best be Conceived by means of the particular cases in which they are surfaces of revolution. Let an ellipse revolve about one of its axes, and let all the circular sections be flattened into ellipses : the result will be an ellipsoid, derived from its particular case, the spheroid. Let an hyper bola revolve about its minor axis; the two branches will generate only one branch of a surface: let the circular sections be flattened into ellipses, and the result is the single hyperboloid. Let the hyperbola revolve about its major axis : the two branches will generate two branches of a surface ; and if the circular sections be flattened into ellipses, the result Is the double hyperboloid. For the elliptic para boloid, let a parabola revolve about its principal axis, and let the circular sections become ellipses. The hyperbolic paraboloid has no surface of revolution among its cases, but its farm may be conceived as follows :—Let two parabolas have a common vertex, and let their planes be at right angles to one another, being turned contrary ways. Let the one parabola then move over the other, always continuing parallel to its first position, and having its vertex constantly on the other : its are will then trace out an hyperbolic paraboloid.
The ellipsoid and the two hyperboloids have centres, but neither of the paraboloids has one. The surfaces which have centres possess an infinite number of triple systems of diameters having properties corre sponding to those of the conjugate diameters of an ellipse and hyper bola. These we shall not describe, but shall proceed to point out how to determine the position of the centre and principal diameters or axes (that is, the system of conjugate diameters, each of which is at right angles to the other two) in either of the surfaces having a centre. Resuming the original equation, and the co-ordinates being supposed rectangular, the co-ordinates of the centre, x, v, and z, are thus deter mined. They are fractions whose denominator is and whose numerators are (be— + (a'b'— (ca—b9b" + (b'c'—aa')c" + (a'b' +cc")a" (ab—cl)c" (e'a'—bb')a" + (6'c'—bb')b" ; and if the origin be removed to the centre, the axes retaining their original directions, the equation of the surface becomes ax: + by: + +2a'yz + 2b'ex + 2c'xy + 0,where w ie the expression already signified by that letter, and will be found to be also xa" + Y6" + zc" +1.
Let the thrce principal axes now make angles with the axes of and e, as follows :—The first, angles whose cosines are a, /3, 7 ; the second, angles whose cosines are a, 13', 7'; the third, tutgles whose cosines are a', ft", 7'. The equation s-v,r—vestO has always three real roots; let them be A, A', Then the directions of the principal axes are to be determined from be— — (/• (A — A) (A —A") /12. ea (c + +