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we may suppose, as in 372, that p is a given vector function of two scalar variables, x and y, between which there will then arise, by the same fundamental formula II., a differential equation of the first order and second degree, to be integrated (when possible) by known methods. For example, if we write,
XL... p = ix + jy + kz, dz=pdx + qdy,
we shall satisfy the equation III. by assuming (with a constant factor understood), XLI... v=ip +jq-k, whence XLII... dv=idp +jdq;
and thus the general equation II., for the lines of curvature on an arbitrary surface, receives (by the laws of ijk) the form,
XLIII... dp (dy + qdz) = dq (dx + pdz);
which last form has accordingly been assigned, and in several important questions employed, by Monge*: but which is now seen to be included in the still more concise (and more easily deduced and interpreted) quaternion equation,
Svdvdp = 0.
411. For a central surface of the second order, we have as usual v = Op, Av = PAP, and therefore (by 347, 348, and by the self-conjugate form of ø),
I... VvAv VøppAp=YVpAp=mp1VpAp;
the general condition of intersection 410, I. of two normals, at the extremities of a finite chord Ap, and the general differential equation 410, II. of the lines of curvature, may therefore for such a surface receive these new and special forms :
forming, by quaternions, the well known equation (d), occurred early to the present writer, and will be briefly mentioned somewhat farther on. In the mean time it may be remarked, that because m” = 0 by (c), when the equation (d) is satisfied, we have then, by the general theory III. ii. 6 of linear and vector functions, and especially by the subarticles to 350, remembering that is here self-conjugate, the formulæ,
(f)... dv + xdp=0, and (g). . . ¥o - p2o = m'o,
X, being auxiliary functions, and m' another coefficient of the cubic, while σ is an arbitrary vector. For the same reason, and under the same condition (d), the function itself has the properties expressed by the equations,
VIK = K$i = 10k, and (i). φ νικ = φιφκ - ηνικ ; in which the two vectors i, k are arbitrary, and m' is the same scalar coefficient as
* See the enunciation of the formula here numbered as XLIII., in page 133 of Liouville's Monge: compare also the applications of it, in pages 274, 303, 305, 357. (The corresponding pages of the Fourth Edition are, 115, 240, 265, 267, 312.) The quaternion equation, Svdvdp = 0, was published by the present writer, in a communication to the Philosophical Magazine, for the month of October, 1847 (page 289). See also the Supplement to the same Volume xxxi. (Third Series); and the Proceedings of the Royal Irish Academy for July, 1846.
CHAP. III.] CONDITION OF INTERSection of normals. 675
II... SAP 'VpAp=0, or II'... SpApp1Ap=0;
III... Sdpp-1Vpdp = 0, or III'... Spdpp'dp = 0;
which admit of geometrical interpretations, and conduct to some new theorems, especially when they are transformed as follows: IV... S\Ap. SpApp1μ + SμAp. SpApp1λ = 0,
V... Sdp. Spdpp1μ+ Sudp.Spdpp1λ = 0,
λ and u being (as in 405, (5.), &c.) the two real cyclic normals of the surface: while the same equations may also be written under the still more simple forms,
VI... SaAp. Sa'pAp + Sa'Ap. Sap▲p = 0,
VII... Sadp. Sa'pdp + Sa'dp. Sapdp = 0,
a, a' being, as in several recent investigations, the two real focal unit lines, which are common to a whole confocal system.
(1.) The vector p-1VpAp in II. has by I. the direction of VvAv; whence, by 410, (6.), the interpretation of the recent equation II., or (for the present purpose) of the more general equation 410, I., is that the chord PP' is perpendicular to its own polar, if the normals at its extremities intersect. Accordingly, if their point of intersection be called N, the polar of PP' is perpendicular at once to PN and P'N, and therefore to PP' itself.
(2.) The equation II'. may be interpreted as expressing, that when the normals at P and P' thus intersect in a point N, there exists a point p" in the diametral plane OPP', at which the normal P'N" is parallel to the chord PP': a result which may be otherwise deduced, from elementary principles of the geometry of surfaces of the second order.
(3.) It is unnecessary to dwell on the converse propositions, that when either of these conditions is satisfied, there is intersection (or parallelism) of the two normals at P and P': or on the corresponding but limiting results, expressed by the equations III. and III'.
(4.) In order, however, to make any use in calculation of these new forms II., III., we must select some suitable expression for the self-conjugate function, and deduce a corresponding expression for the inverse function 1. The form,*
VIII. . . φρ = gp + Πλρμ,
which has already several times occurred, has also been more than once inverted : but the following new inverse† form,
*The vector form VIII. occurred, for instance, in pages 463, 469, 474, 484, 641, 669; and the connected scalar form,
+ Inverse forms, for p1p or m1p, have occurred in pages 463, 484, 641 (the
has an advantage, for our present purpose, over those assigned before. In fact, this form IX. gives at once the equation,
X. . . (9 – Sλμ). p ̈1Vp▲p=Vp▲p—\SpApp-1μ – μSpApo ̄1X ;
and so conducts immediately from II. to IV., or from III. to V. as a limit.
(5.) The equation IV. expresses generally, that the chord Ap, or PP', is a side of a certain cone of the second order, which has its vertex at the point P of the given surface, and passes through all the points p' for which the normals to that surface intersect the given normal at P; and the equation V. expresses generally, that the two sides of this last cone, in which it is cut by the given tangent plane at the same point P, are the tangents to the lines of curvature.
(6.) But if the surface be an ellipsoid, or a double-sheeted hyperboloid, then (comp. 408, (29.)) the always real vectors,* -1X and p ̄1μ, have the directions of semidiameters drawn to two of the four real umbilics; supposing then that p is such a semidiameter, and that it has the direction of +-1A, the second term of the first member of the equation IV. vanishes, and the cone IV. breaks up into a pair of planes, of which the equations in p' are,
XI. . . S\ (p'− p)=0, and XII. . . Sp'-1λ-1μ = 0;
whereof the former represents the tangent plane at the umbilic r, and the latter represents the plane of the four real umbilics.
(7.) It follows, then, that the normal at the real umbilic P is not intersected by any real normal to the surface, except those which are drawn at points P' of that principal section, on which all the real umbilics are situated: but that the same real umbilicar normal PN is, in an imaginary sense, intersected by all the imaginary normals, which are drawn from the imaginary points P' of either of the two imaginary generatrices through P.
(8.) In fact, the locus of the point P', under the condition of intersection of its normal P'N' with a given normal PN, is generally a quartic curve, namely the intersection of the given surface with the cone IV.; but when this cone breaks up, as in (6.), into two planes, whereof one is normal, and the other tangential to the surface, the general quartic is likewise decomposed, and becomes a system of a real conic, namely the principal section (7.), and a pair of imaginary right lines, namely the two umbilicar generatrices at P.
(9.) We see, at the same time, in a new way (comp. 410, (14.)), that each such generatrix is (in an imaginary sense) a line of curvature: because the (imaginary) normals to the surface, at all the points of that generatrix, are situated by (7.) in one common (imaginary) normal plane.
(10.) Hence through a real umbilic, on a surface of the second order, there pass
correction in a Note to which last page should be attended to). In comparing these with the form IX., it will easily be seen (comp. page 661) that
CHAP. III.] THREE LINES THROUGH AN UMBILIC.
three lines of curvature: whereof one is a real conic (8.), and the two others are imaginary right lines, namely, the umbilicar generatrices as before.
(11.) If we prefer differentials to differences, and therefore use the equation V. of the lines of curvature, we find that this equation takes the form 0=0, if the point P be an umbilic; and that if the normal at that point be parallel to A, the differential of the equation V. breaks up into two factors, namely,
XIII... Sλd2p = 0, and XIV... Sdpp ̄1λp ̄1μ = 0;
whereof the former gives two imaginary directions, and the latter gives one real direction, coinciding precisely with the three directions (10.).
(12.) And if p, instead of being the vector of an umbilic, be only the vector of a point on a generatrix corresponding, we shall still satisfy the differential equation V., by supposing that dp belongs to the same imaginary right line: because we shall then have, as at the umbilic itself,
XV... SXdp = 0, Spdpp-1λ = 0.
An umbilicar generatrix is therefore proved anew (comp. (9.)) to be, in its whole extent, a line of curvature.
(13.) The recent reasonings and calculations apply (6.), not only to an ellipsoid, but also to a double-sheeted hyperboloid, four umbilics for each of these two surfaces being real. But if for a moment we now consider specially the case of an ellip
a + c
soid, and if we denote for abridgment the real quotient by h, we may then substitute in IV. and V. for λ, μ, -1, 4-1μ the expressions,
ha - a' =
a + c
ac (a + c)
and then, after division by h2 1, there remain only the two vector constants a a', the equation IV. reducing itself to VI., and V. to VII.
(14.) The simplified equations thus obtained are not however peculiar to ellipsoids, but extend to a whole confocal system. To prove this, we have only to combine the equations II. and III. with the inverse form,
XVIII. . . l-2 ̄'p = aSa′p + a’Sap − p(e + Saa'),
which follows from 407, XV., and gives at once the equations VI. and VII., whatever the species of the surface may be.
(15.) The differential equation VII. must then be satisfied by the three rectangular directions of dp, or of a tangent to a line of curvature, which answer to the orthogonal intersections (410, (12.)) of the three confocals through a given point P ; it ought therefore, as a verification, to be satisfied also, when we substitute v for dp, being a normal to a confocal through that point: that is, we ought to have the equation,
XIX... SavSa'pv + Sa′vSapv = 0).
And accordingly this is at once obtained from 407, XVI., by operating with S.pv; so that the three normals v are all sides of this cone XIX., or of the cone VII. with dp for a side, with which the cone V. is found to coincide (13.).
(16.) And because this last equation XIX., like VI. and VII., involves only the two focal lines a, a' as its constants, we may infer from it this theorem: "If inde
finitely many surfaces of the second order have only their asymptotic cones biconfocal,* and pass through a given point, their normals at that point have a cone of the second order for their locus;" which latter cone is also the locus of the tangents, at the same point, to all the lines of curvature which pass through it, when different values are successively assigned to the scalar constant a2 – c2 (or 2/3): that is, when the asymptotes a, a' to the focal hyperbola remain unchanged in position, but the semiaxes (a2-b2), (b2 — c2) of that curve (here treated as both real) vary together.
(17.) The equation VI. of the cone of chords (5.) introduces the fixed focal lines a, a' by their directions only. But if we suppose that the lengths of those two lines are equal, without being here obliged to assume that each of those lengths is unity, we shall then have (comp. 407, (2.), (3.)), the following rectangular system of unit lines, in the directions of the axes of the system,
XX... U (a + a'), UVaa', U(a−a'), '
which obey in all respects the laws of ijk, and may often be conveniently denoted by those symbols, in investigations such as the present. And then, by decomposing the semidiameter p, and the chord Ap, in these three directions XX., we easily find the following rectangular transformation† of the foregoing equation VI.,
in which it is permitted to change Ap to do, in order to obtain a new form of the differential equation of the lines of curvature; or else at pleasure to v, and so to find, in a new way, a condition satisfied by the three normals, to the three confocals through P.
(18.) The cone, VI. or XXI., is generally the locus of a system of three rectangular lines; each plane through the vertex, which is perpendicular to any real side, cutting it in a real pair of mutually rectangular sides: while, for the same reason, the section of the same cone, by any plane which does not pass through its vertex P, but cuts any side perpendicularly, is generally an equilateral hyperbola.
(19.) If, however, the point r be situated in any one of the three principal planes, perpendicular to the three lines XX., then the cone XXI. (as its equation shows) breaks up (comp. (6.)) into a pair of planes, of which one is that principal
*That is, if the surfaces (supposed to have a common centre) be cut by the plane at infinity in biconfocal conics, real or imaginary.
†The corresponding form, in rectangular co-ordinates, of the condition of intersection, of normals at two points (xyz) and (x'y'z′), to the surface,
is the equation (probably a known one, although the writer has not happened to meet with it),
in which it is evident that xyz and x'y'z' may be interchanged.