maginary right lines is seen anew to be a line* of curvature, on the sur- hee (comp. 410, (h)), because all the normals P'N', at points of this e are situated in one common (imaginary) normal plane (p. 676): and as before, there are thus three lines of curvature through an um- (f). These geometrical results are in various ways deducible from calculation with quaternions; for example, a form of the equation of the lines of curvature on a quadric is seen (p. 677) to become an identity at an umbilic (A): while the differential of that equation breaks up into two factors, whereof one represents the tangent to the principal section, while the other (Sλd2p = 0) assigns the directions of (s). The equation of the cone (C), which has already presented itself as a certain locus of chords (b), admits of many quaternion Sa Ap Sa Ap = 0, ℗ being the vector of the vertex P, and p + Ap that of any other point (A). This cone (C), or (V1), is also the locus (p. 678) of a system * It might be natural to suppose, from the known general theory (410, (c)) of the two rectangular directions, that each such generatrix PP' is crossed perpendi- cularly, at every one of its non-umbilicar points r', by a second (and distinct, although imaginary) line of curvature. But it is an almost equally well known and received result of modern geometry, paradoxical as it must at first appear, that when a right line is directed to the circle at infinity, as (by 408, (e)) the gene- ratrices in question are, then this imaginary line is everywhere perpendicular to itself. Compare the Notes to pages 459, 672. Quaternions are not at all re- sponsible for the introduction of this principle into geometry, but they recognise and employ it, under the following very simple form: that if a non-evanescent vector be directed to the circle at infinity, it is an imaginary value of the symbol 01 (comp. pp. 300, 459, 662, 671, 672); and conversely, that when this last symbol represents a vector which is not null, the vector thus denoted is an imaginary line, which cuts that circle. It may be noted here, that such is the case with the reci- procal polar of every chord of a quadric, connecting any two umbilics which are not in one principal plane'; and that thus the quadratic equation (XXI., in p. 669) from which the two directions (410, (c)) can usually be derived, becomes an iden- tity for every umbilic, real or imaginary: as it ought to do, for consistency with the foregoing theory of the three lines through that umbilic. And as an addi- tional illustration of the coincidence of directions of the lines of curvature at any non-umbilicar point p' of an umbilicar generatrix, it may be added that the cone of chords (0), in 411, (b), is found to touch the quadric along that generatrix, f of three rectangular lines; and if it be cut by any plane perpendicular to a side, and not passing through the vertex, the section is an equila- (i). The same cone (C) has, for three of its sides PP', the normals (p. 677) to the three confocals (p. 644) of a given system which pass (j). And because its equation (V1) does not involve the constant l, of 407, (a), (b), we arrive at the following theorem (p. 678):-Iƒ indefinitely many quadrics, with a common centre o, have their asymp totic cones biconfocal, and pass through a common point P, their normals (a). If a be the vector of the centre s of curvature of a normal section of an arbitrary surface, which touches one of the two lines of curvature thereon, at any given point P, we have the two fundamental the equation (Wi′′) being a new form of the general differential equa- (b). Deduction (pp. 680, 681, &c.) of some known theorems from (c). Introducing the auxiliary scalar (p. 682), in which 7 (|| dp) is a tangent to a line of curvature, while dv = pdp, as in (U1), the two values of r, which answer to the two rectangular directions (T1") in 410, (c), are given (p. 680) by the expression, in which g, λ, μ are, for any given point P, the constants in the equa- (d). At any other point P of the given surface, which is as yet en- tirely arbitrary, the values of r may be thus expressed (p. 681), a1, a2 being the scalar semiaxes (real or imaginary) of the index curve CONTENTS. (e). The quadratic equation, of which r1 and r1⁄2, or the inverse Sv ̄1 (p+r) ̄1v=0; which may be developed (same page) into this other form, (Y1) the linear and vector functions, and x, being derived from the func- tion ø, on the plan of the Section III. ii. 6 (pp. 440, 443). (f). Hence, generally, the product of the two curvatures of a sur- which will be found useful in the following series (413), in connexion with the theory of the Measure of Curvature. (9). The given surface being still quite general, if we write = Udp, r' = U (vdp), (A2), and therefore rr'= Uv, (A2) this general parallelism of dr' to r being geometrically explained, by (h). If the vector of curvature (389) of a line of curvature be (1). The vector σ of a centre s of curvature of the given surface, σ =p+r1v; (C2) which may be regarded also as a general form of the Vector Equation (j). The normal at s, to what may be called the First Sheet, has corresponding to s, on the corresponding sheet of the Reciprocal (comp. pp. 507, 508) of the Surface of Centres, has (by p. 684) the expres which may also be considered (comp. (i)) to be a form of the Vector Equation of that Reciprocal Surface. (k). The vector v satisfies generally (by same page) the equations do, du denoting any infinitesimal variations of the vectors σ and v, consistent with the equations of the surface of centres and its recipro- Spv=1, Svv=0, Svvpv = 0. (D2") in which w is a variable vector, represents (p. 684) the normal plane to the first line (j) of curvature at P; or the tangent plane at s to the first sheet of the surface of centres: or finally, the tangent plane to that developable normal surface (g), which rests upon the second line of curvature, and touches the first sheet along a certain curve, whereof we shall shortly meet with an example. And if v be regarded, comp. (i), as a vector function of two scalar variables, the envelope of the variable plane (E2) is a sheet of the surface of centres; or rather, on account of the ambiguous sign (i), it is that surface of centres itself: while, in like manner, the reciprocal surface (j) is the envelope of this other (m). The equations (W1), (W1') give (comp. the Note to p. 684), combining which with (C2), we see that the equations (H1) of p. xxv. (n). In the foregoing paragraphs of this analysis, the given sur- face has throughout been arbitrary, or general, as stated in (d) and (9). But if we now consider specially the case of a central quadric, several less general but interesting results arise, whereof many, but perhaps not all, are known; and of which some may be mentioned Pages. CONTENTS. (o). Supposing, then, that not only dv=ødp, but also v=pp, and Spv=fp=1, the Index Surface (410, (d)) becomes simply (p. 670) the given surface, with its centre transported from o to P; whence many simplifications follow. (p). For example, the semiaxes a1, as of the index curve are now equal (p. 681) to the semiaxes of the diametral section of the given surface, made by a plane parallel to the tangent plane; and Tv is, as in 409, the reciprocal P-1 of the perpendicular, from the centre on this latter plane; whence (by (X1) and Xi")) these known expressions for the two curvatures result: (q). Hence, by (e), if a new surface be derived from a given central quadric (of any species), as the locus of the extremities of normals, erected at the centre, to the planes of diametral sections of the given surface, each such normal (when real) having the length of one of the semiazes of that section, the equation of this new surfacet admits (p. 683) of being written thus: (r). Under the conditions (o), the expression (C2) for a gives (p. 684) the two converse forms, xxxvii Pages. and therefore (p. 689), by (d), (p), and by the theory (407) of confocal surfaces, if 2 be formed from by changing the semiaxes abe to a2b2c2; it being understood that the given quadric (abe) is cut by the two confocals (abic) and (a2b2c2), in the first and second lines of curvature through the given point P: and that σ is here the vector of that first centre s of curvature, which answers to the first line (comp. (j). Of course, on the same plan, we have the analogous expression, Throughout the present Series 412, we attend only (comp. (a)) to the curvatures of the two normal sections of a surface, which have the directions of the two lines of curvature: these being in fact what are always regarded as the two principal curvatures (or simply as the two curvatures) of the surface. But, in a shortly subsequent Series (414), the more general case will be considered, of the curvature of any section, normal or oblique. + When the given surface is an ellipsoid, the derived surface is the celebrated Wave Surface of Fresnel: which thus has (H2) for a symbolical form of its equation. When the given surface is an hyperboloid, and a semiaxis of a section is imaginary, the (scalar and now positive) square, of the (imaginary) normal erected, is still to be made equal to the square of that semiaxis. |