CHAPTER VIII. SURFACES OF THE SECOND ORDER. 248. THE HE general scalar equation of the second order in a vector p must evidently contain a term independent of p, terms of the form S.apb involving p to the first degree, and others of the form S.apbpc involving p to the second degree, a, b, c, &c. being constant quaternions. Now the term S.apb may be written each of which may evidently be put in the form Syp, where is a known vector. Similarly the term S.apbpc may be reduced to a set of terms, each of which has one of the forms Ap2, (Sap)', Sap SBP, the second being merely a particular case of the third. Thus (the numerical factors 2 being introduced for convenience) we may write the general scalar equation of the second degree as follows:: 22. Sap SBp+ Ap2 + 2 Syp = C. ..... (1) 249. Change the origin to D where OD 8, then p becomes P+d, and the equation takes the form 22. Sap SBp+ Ap2 + 2 Σ (Sap Sẞò + Sẞp Sad) + 2 A Sồp + 2 Syp +22. Sad Sẞò + Ad2 + 2 Syd — C = 0; from which the first power of p disappears, that is the surface is referred to its centre, if Σ(aSẞd+ẞSad) + Aò + y = 0, (2) a vector equation of the first degree, which in general gives a single definite value for 8, by the processes of Chapter V. [It would lead us beyond the limits of an elementary treatise to consider the special cases in which (2) represents a line, or a plane, any point of which is a centre of the surface. The processes to be employed in such special cases have been amply illustrated in the Chapter referred to.] With this value of 8, and putting DC-2 Syd - Ad2 — 2 Σ. Sad Sßd, the equation becomes 22. Sap SBp+Ap2 = D. If D=0, the surface is conical (a case treated in last Chapter); if not, it is an ellipsoid or hyperboloid. Unless expressly stated not to be, the surface will, when D is not zero, be considered an ellipsoid. By this we avoid for the time some rather delicate considerations. By dividing by D, and thus altering only the tensors of the constants, we see that the equation of central surfaces of the second order, referred to the centre, is (excluding cones) 22(Sap Sẞp)+gp2 = 1. 250. Differentiating, we obtain or ..... 22{Sadp SBp+ Sap Sẞdp}+2g Spdp and therefore, by § 137, the tangent plane is i. e. Hence if S(w-p) {(aSBp+B Sap)+gp} = 0, = 0, S. {(a SBp+B Sap)+gp} = 1, by (3). v = Σ(aSßp+B Sap)+gp (3) (4) And, as v1 is evidently the vector-perpendicular from the origin on the tangent plane, v is called the vector of proximity. 251. Hamilton uses for v, which is obviously a linear and vector function of p, the notation op, expressing a functional operation, as in Chapter V. But, for the sake of clearness, we will go over part of the ground again, especially for the benefit of students who have mastered only the more elementary parts of that Chapter. (a.) (p+o) = op+po, &c., for any two or more vectors. (d.) Soop = (Sao SBp+Sẞo Sap)+g Spo = Sppo, or is, in this case, self-conjugate. This last property is of great importance. 252. Thus the general equation of central surfaces of the second degree (excluding cones) may now be written Spop=1. .... (1) Differentiating, Sdpop+Spdop = 0, which, by applying (c.) and then (d.) to the last term on the left, gives 2 Sppdp=0, and therefore, as in § 250, though now much more simply, the tangent plane at the extremity of p is This is therefore the equation of the plane of contact of tangent planes drawn from 4. 253. To find the enveloping cone whose vertex is 4, notice that (Spop-1)+p(Sppa-1)2 = 0, where p is any scalar, is the equation of a surface of the second order touching the ellipsoid along its intersection with the plane. If this pass through we have (Sapa-1)+p(Sapa — 1)2 = 0, and p is found. Then our equation becomes (Sppp-1) (Sapa-1)-(Sppa-1)2 = 0, which is the cone required. To assure ourselves of this, transfer the origin to A, by putting p+a for The result is, using (a.) and (d.), ρ. (Sppp +2 Sppa + Sapa−1)(Sapa−1)—(Sp‡a+Sapa — 1)2 = 0, which is homogeneous in Tp2, and is therefore the equation of a cone. Suppose A infinitely distant, then we may put in (1) ra for a, where x is infinitely great, and, omitting all but the higher terms, the equation of the cylinder formed by tangent lines parallel to a is 254. To study the nature of the surface more closely, let us find the locus of the middle points of a system of parallel chords. Let them be parallel to a, then, if be the vector of the middle point of one of them, + xa and values of p which ought to satisfy (1) of § 252. That is -xa are simultaneous S.(w + xa) $ (≈ + xa) = 1. Hence, by (a.) and (d.), as before, Sapa+Sapa = 1, (1) The latter equation shows that the locus of the extremity of, the middle point of a chord parallel to a, is a plane through the centre, whose normal is pa; that is, a plane parallel to the tangent plane at the point where OA cuts the surface. And (d.) shows that this relation is reciprocal-so that if ẞ be any value of a, i. e. be any vector in the plane (1), a will be a vector in a diametral plane which bisects all chords parallel to B. The equations of these planes are = Sapa = 0, δαφβ: so that if V.papẞy (suppose) is their line of intersection, we have Sypa =0= Sapy and (1) gives Hence there is Sẞpa = 0 = an infinite number of sets of three vectors a, ß, y, such that all chords parallel to any one are bisected by the diametral plane containing the other two. 255. It is evident from § 23 that any vector may be expressed as a linear function of any three others not in the same plane, let then so that a, ẞ, and y are vector conjugate semi-diameters of the surface we are engaged on. Substituting the above value of p in the equation of the |