CHAP. III.] INDEX CURVE AND SURFACE, CONJ. TANGENTS. 669

(6.) The parallelism XII'. may be otherwise expressed by saying (comp. (4.)) that

XIV... do and Vvdv

have the directions of conjugate tangents; or that the two vectors,

XV... Δρ and Vvv,

have ultimately such directions, when Tap diminishes indefinitely. But whatever may be this length of the chord Ap, the vector Vvv has the direction of the line of intersection of the two tangent planes to the surface, drawn at its two extremities: another theorem of Dupin* is therefore reproduced, namely, that if a developable be circumscribed to any surface, along any proposed curve thereon, the generating lines of this developable are everywhere conjugate, as tangents to the surface, to the corresponding tangents to the curve, with the recent definition (4.) of such conjugation.

(7.) The following is a very simple mode of proving by quaternions, that if a tangent satisfies the equation VI., then the rectangular tangent,

XVI. . . τ' = ντ,

satisfies the same equation. For this purpose we have only to observe, that the selfconjugate property of gives, by VI. and XVI.,

XVII. . . 0 = Στ'φτ = Sror' = v-2Ѕντ'φτ'.

(8.) Another way of exhibiting, by quaternions, the mutual rectangularity of the lines of curvature, is by employing (comp. 357, I.) the self-conjugate form, XVIII. . . φτ = gr + Vλτμ ;

in which the vectors λ, μ, and the scalar g, depend only on the surface and the point, and are independent of the direction of the tangent. The equation VI. then becomes by V.,

ΧΙΧ. . . 0 = $vrr = SvrSur + Sντμλr;

..

ΧΧΙ. . . y2 (Vνμ)2 = x2 (Vvd), or XXI...yTV vμ = + xTVνλ ;

the two directions of rare therefore those of the two lines,

ΧΧΙΙ. . . UVυλ + UVνμ,

which are evidently perpendiculart to each other.

* Dupin proved first (Dév. de Géométrie, pp. 43, 44, &c.), that two such tangents as are described in the text have a relation of reciprocity to each other, on which account he called them "tangentes conjuguées:" and afterwards he gave a sort of image, or construction, of this relation and of others connected with it, by means of the curve which he named "l'indicatrice" (in his already cited page 48, &c.).

† This mode, however, of determining generally the directions of the lines of curvature, gives only an illusory result, when the normal has the direction of either A or μ, which happens at an umbilic of the surface. Compare 408, (27.), (29.), and the first Note to page 466.

(9.) An interpretation, of some interest, may be given to this last expression XXII., by the introduction of a certain auxiliary surface of the second order, which may be called the Index Surface, because the index curve (4.) is the diametral section of this new surface, made by the tangent plane to the given one. With the recent signification of 6, this index surface is represented by the equation VII., if be now supposed (comp. (2.)) to represent a line pr drawn in any direction from the given point P, and therefore not now obliged to satisfy the condition V. of tangency. Or if, for greater clearness, we denote by p + p' the vector from the origin o to a point of the index surface, the equation to be satisfied is, by the form XVIII. of $ (comp. 357, II.),

ΧΧΙΙΙ... 1 = Sp ́φρ' =gp + βλρ'μρ ́;

the centre of this auxiliary surface being thus at P, and its two (real) cyclic normals being the lines A and u: so that Vvd and Vυμ have the directions of the traces of its two cyclic planes, on that diametral plane (Svp' = 0) which touches the given surface. We have therefore, by XXII., this general theorem, that the bisectors of the angle formed by these two traces are the tangents to the two lines of curvature, whatever the form of the given surface may be.

(10.) Supposing now that the given surface is itself one of the second order, and that its centre is at the origin o, so that it may be represented (comp. 405, ΧΙΙ.) by the equation,

XXIV... 1 = Spop = gp3 + Sλρμρ,

with constant values of λ, μ, and g, which will reproduce with those values the form XVIII. of o, we see that the index surface (9.) becomes in this case simply that given one, with its centre transported from o to P; and therefore with a tangent plane at the origin, which is parallel to the given tangent plane. And thus the traces (9.), of the cyclic planes on the diametral plane of the index surface, become here the tangents to the circular sections of the given surface. We recover then, as a case of the general theorem in (9.), this known but less general theorem: that the angles formed by the two circular sections, at any point of a surface of the second order, are bisected by the lines of curvature, which pass through the same point.

(11.) And because the tangents to these latter lines coincide generally, by (3.) (4.) (9.), with the axes of the diametral section of the index surface, made by the tangent plane to the given surface, they are parallel, in the case (10.), as indeed is well known, to the axes of the parallel section of a given surface of the second

order:

(12.) And if we now look back to the Equation of Confocals in 407, (26.), and to the earlier formule of 407, (4.), we shall see that because the vector vi, in the last cited sub-article, represents a tangent to the given surface Spop = 1, complanar* with the normal v and the derived vector øv, so that it satisfies (comp. 407, XII. XIV., and the recent formulæ V. VI.) the two scalar equations,

XXV... Svv1 = 0, and XXVI... Σννιφνι = 0,

which are likewise satisfied (comp. (7.)) when we change vi to the rectangular tan

* Compare the Note to page 645.

CHAP. III.] LINES OF CURVATURE ON CENTRAL SURFACES. 671

gent v2, it follows that these two vectors, v1 and v2, which are the normals to the two confocals to (e) through P, are also the tangents to the two lines of curvature on that given surface of the second order at that point: whence follows this other theorem* of Dupin, that the curve of orthogonal intersection (407, (4.)), of two confocal surfaces, is a line of curvature on each.

(13.) And by combining this known theorem, with what was lately shown respecting the umbilicar generatrices (in 408, (30.), (32.), comp. also (35.), (36.)), we may see that while, on the one hand, the lines of curvature on a central surface of the second order have no real envelope, yet on the other hand, in an imaginary sense, they have for their common envelopet the system of the eight imaginary right lines (408, (31.)), which connect the twelve (real or imaginary) umbilics of the surface, three by three, and are at once generating lines of the surface itself, and also of the known developable envelope of the confocal system.

(14.) It may be added, as another curious property of these eight imaginary right lines, that each is, in an imaginary sense, itself a line of curvature upon the surface: or rather, each represents two coincident lines of that kind. In fact, if we denote the variable vector 408, LXXX. of such a generatrix by the expression,

XXVII. . . ρ = ε'σ + σ',

in which e' is a variable scalar, but σ, σ' are two given or constant but imaginary vectors, such that

and

XXVIII... σ2 = 0, 8σσ ́=-12, 62 = - 62,

XXIX... fo = $σφσ = 0, f(σ, σ') = $σ'φσ = 0, fo' = 1,

we have the imaginary normal v, with (for the case of a real umbilic) a real tensor,

ΧΧΧ. . . ν = ε'φσ + φσ ́ 1 σ, XXXI... Tv = +

(e-e') 12
abc

;

* Dev. de Géométrie, page 271, &c.

† The writer is not aware that this theorem, to which he was conducted by quaternions, has been enunciated before; but it has evidently an intimate connexion with a result of Professor Michael Roberts, cited in page 290 of Dr. Salmon's Treatise, respecting the imaginary geodetic tangents to a line of curvature, drawn from an umbilicar point, which are analogous to the imaginary tangents to a plane conic, drawn from a focus of that curve. An illustration, which is almost a visible representation, of the theorem (13.) is supplied by Plate II. to Liouville's Monge (and by the corresponding plate in an earlier edition), in which the prolonged and dotted parts of certain ellipses, answering to the real projections of imaginary portions of the lines of curvature of the ellipsoid, are seen to touch a system of four real right lines, namely the projections (on the same plane of the greatest and least axes), of the four real umbilicar tangent planes, and therefore also of what have been above called (408, (30.), (31.)) the eight (imaginary) umbilicar generatrices of the surface, Accordingly Monge observes (page 150 of Liouville's edition), that "toutes les ellipses, projections des lignes de courbure, seront inscrites dans ce parallélogramme dont chacune d'elles touchera les quatre côtés :" with a similar remark in his explanation of the corresponding Figure (page 160).

and we find, after reductions, the imaginary expression,

ΧΧΧΙΙ. . . νσ = + V-1 στν, whence XXXIII... Svo = 0, δυσφσ = 0. The differential equations V. VI. of a line of curvature are therefore symbolically satisfied, when we substitute, for the tangential vector r, either the imaginary line itself, or the apparently perpendicular but in an imaginary sense coincident vector vo; and the recent assertions are justified.

(15.) As regards the real lines of curvature, on a central surface of the second order, we see by comparing the general differential equation II. with the expression 409, XXIII. for the differential of h, or of P-2D-3, that this latter product, or the product P. D itself, is constant for a line of curvature, as well as for a geodetic line, on such a surface, as indeed it is well known to be: although this last constant (P. D) may become imaginary, for the case of a single-sheeted hyperboloid, and must be such for a line of curvature on an hyperboloid of two sheets.

(16.) And as regards the general theory of the index surface (9.), it is to be observed that this auxiliary surface depends primarily on the scalar function f, in the equation fp = 1, or generally fo = const., of the given surface; and that it is not entirely determined by means of that surface alone. For if we write, for instance,

XXXIV...ffp=f1, with dfp = 2Svdp as before,

we shall have, as the new first differential equation of the same given surface, instead of III.,

XXXV...0= dffp = 2Sηνdρ, with XXXVI... n = ffp;

and if we then write, by analogy to IV.,

XXXVII...d. nv = όλρ = ηφάρ + n'vSvdp, with XXXVIII. . . n' = 2f"fp, the new index surface, constructed on the plan (9.), will have for its equation, analogous to XXIII., the following:

ΧΧΧΙΧ... Αρ'φρ' = ηρ'φρ' + n' (Svp')3 = const.

* As regards the paradox, of the imaginary vector o being thus apparently perpendicular to itself, a similar one had occurred before, in the investigation 353, (17.), (18.), (19.); and it is explained, on the principles of modern geometry, by observing that this imaginary vector is directed to the circle at infinity. Compare 408, (31.), and the Note to page 459.

† Compare the first Note to page 667.

Although the writer has been content to employ, in the present work, some of these usual but rather long appellations, he feels the elegance of Dupin's phraseology, adopted also by Möbius, and by some other authors, according to which the two central hyperboloids are distinguished, as elliptic (for the case of two sheets), and hyperbolic (for the case of one). The phrase "quadric," for the general surface of the second order (or second degree), employed by Dr. Salmon and Mr. Cayley, is also very convenient. It may be here remarked, that Dupin was perfectly aware of, or rather appears to have first discovered, the existence of what have since his time come to be called the focal conics; which important curves were considered by him, as being at once limits of confocal surfaces, and also loci of umbilics. Comp. Dév. de Géométrie, pages 270, 277, 278, 279; see also page 390 of the Aperçu Historique, &c., by M. Chasles (Brussels, 1837).

CHAP. III.] FORMS OF DIFFERENTIAL EQUATION.

673

(17.) But if we take this last constant = n, the two index surfaces, XXIII. and XXXIX., will have a common diametral section, made by the given tangent plane, namely the index curve (4.); and they will touch each other, in the whole extent of that curve. And it will be found that the construction (9.), for the directions of the lines of curvature, applies equally well to the one as to the other, of these two auxiliary surfaces: in fact, it is evident that the differential equation II., namely Svdvdp = 0, receives no real alteration, when vis multiplied by any scalar, n, even if that scalar should be variable.

(18.) And instead of supposing that the variable vector p is thus obliged, as in 373, to satisfy a given scalar equation, of the form*

fp = const.,

* If p = ix + jy + kz, and v=fp = F(x, y, z), and if we write,

dq=q'dy + p"dz + r"dæ, dr = r'dz + q"dx+p"dy,

we may then write also, on the present plan, which gives dfp = 2Svdp,

dv = - (idp+jdq+kdr), Sdpdv = (dxdp + dydq + dzdr);

and the index surface, constructed as in (9.), and with p' changed to Ap = iAx + jay + Az, will thus have the equation,

(a)...p'Ar2 + q'Ay2 + rAz2 + p"AyAz + q"AzAx + r"AxAy = 1,

or more generally = const.; so that it may be made in this way to depend upon, and be entirely determined by, the six partial differential coefficients of the second order, p'..p".., of the function v or fp, taken with respect to the three rectangular coordinates, xyz. And by comparing this equation (a) with the following equation of the same auxiliary surface, which results more directly from the principles employed in the text (comp. XVIII. XXIII.),

(b)... SAρφΔρ = 9Δρ2 + ΒλΔρμΔρ = 1,

we can easily deduce expressions for those six partial coeficients, in terms of g, λ, μ. Thus, for example,

but

D2v=p' = -g + Skipi = Sau-g+ 2SixSiu;
SixSiu + SjxSjp + SkxSkp = - Sλμ; therefore,
(c)...(D+D,20 + D220) = Sλμ - 39 = C1+C2 + C3 = -m",

if c1, C2, C3 be the roots and m" a coefficient of a certain cubic (354, III.), deduced from the linear and vector function dv = odp, on a plan already explained. If then the function v satisfy, as in several physical questions, the partial differential equation,

(d)... Dv + Dy20 + D220 = 0,

the sum of these three roots, C1, C2, C3, will vanish: and consequently, the asymptotic cone to the index-surface, found by changing 1 to 0 in the second member of (a), is real, and has (comp. 406, XXI., XXIX.) the property that

(e)... cota + cot2 b = 1,

if a, b denote its two extreme semiangles. An entirely different method of trans