CHAP. III.]

CENTRES OF CURVATURE.

679

plane itself, while the other is perpendicular thereto. And while the former plane cuts the surface in a principal section, which is always a line of curvature through P, the latter plane usually cuts the surface in another conic, which crosses the former section at right angles, and gives the direction of the second line of curvature.

(20.) But if we further suppose, as in (6.), that the point is an umbilic, then (as has been seen) the second plane is a tangent plane; and the second conic (19.) is itself decomposed, into a pair of imaginary right lines: namely, as before, the two umbilicar generatrices through the point, which have been shown to be, in an imaginary sense, both lines of curvature themselves, and also a portion of the envelope of all the others.

(21.) We shall only here add, as another transformation of the general equation VI. of the cone of chords, which does not even assume Ta = Ta', the following: XXII... S(a + α ́) Δρ. δ (α + α ́) ρΔρ = S(α - α ́) Δρ. (α – α ́) ρΔρ; where the directions of the two new lines, a + a' and a-a', are only obliged to be harmonically conjugate with respect to the directions of the fixed focal lines of the system: or in other words, are those of any two conjugate semidiameters of the focal hyperbola.

412. The subject of Lines of Curvature receives of course an additional illustration, when it is combined with the known conception of the corresponding Centres of Curvature. Without yet entering on the general theory of the curvatures of sections of an arbitrary surface, we may at least consider here the curvatures of those normal sections, which touch at any given point the lines of curvature. Denoting then by the vector of the centre s of curvature of such a section, and by R the radius ps, considered as a scalar which is positive when it has the direction of + v, it is easy to see that we have the two fundamental equations :

whence follows this new form of the general differential equation

410, II. of the lines of curvature,

III... VdpdUv = 0;

with several other combinations or transformations, among which

the following may be noticed here:

(1.) The equation I. requires no proof; and from it the equation II. is obtained by merely differentiating* as if & and R were constant: after which the formula III. follows at once, and IV. is easily deduced.

* To students who are accustomed to infinitesimals, the easiest way is here to

(2.) To obtain from this last equation a more developed expression for R, we may assume for dv, considered as a linear and self-conjugate function of do (410, (1.)), the general form (comp. 410, XVIII.),

V... dv = gdp + Γλάρμ,

in which g, λ, μ are independent of do; and then, while the tangent do has (by 410, XXII.) one or other of the two directions,

VI... dp || UVA + UVνμ,

the curvature R-1 receives one or other of the two values corresponding,

VII... R1 = -Tv-1 (g + SAUv. SuUv + TVAU.TVUν).

(3.) One mode of arriving at this last transformation, or of showing that if (comp. again 410, XXII.) we assume,

then

or

VIII...r= (or ||) UVAv + UV μν,

ΙΧ. . . Αλτμτ-! = SAU. SUv + TVAUν. TVμυν,
Χ... 28λτ. 8μr-1=S(VAUv.VUv) + TVAU. TVμυν,

or finally, XI... 2SUλr. SUμr-1 = S(VUλν. VUμν) + TVUxv. TVUμν,

is to introduce the auxiliary quaternion,

and to prove that, with the value (or direction) VIII. of r, we have thus the equa

tion (in which Vq2, as usual, represents the square of Vq),

ΧΙΙΙ... 2SUAr. SUμr-1=Sq+Tg= Sq+Tg

(4.) And this may be done, by simply observing that we have thus (with the value VIII.) the expressions,

(5.) Admitting then the expression VII., for the curvature R1, we easily sea

and that the difference of the two (principal) curvatures, of normal sections of an arbitrary surface, answering generally to the two (rectangular) directions of the

conceive the differentials to be such. But it has already been abundantly shown, that this view of the latter is by no means necessary, in the treatment of them by quaternions. (Compare the second Note to page 667.)

CHAP. II.]

DIFFERENCE OF CURVATURES.

681

lines of curvature through the particular point considered, vanishes when the normal v has the direction of either of the two cyclic normals, λ, μ, of the index surface (410, (9.)); that is, when the index curve (410, (4.)), considered as a section of that index surface, is a circle: or finally, when the point in question is, in a received sense, an umbilic* of the given surface.

(6.) That surface, although considered to be a given one, has hitherto (in these last sub-articles) been treated as quite general. But if we now suppose it to be a central surface of the second order, and to be represented by the equation,

ΧΙΧ... fp = gp2 + SAρμρ = 1,

which has already several times occurred, we see at once, from the formula VII. or XVIII. (comp. 410, (10.)), that the difference of curvatures, of the two principal normal sections of any such surface, varies proportionally to the perpendicular (Tv-1 or P) from the centre on the tangent plane, multiplied by the product of the sines of the inclinations of that plane, to the two cyclic planes of the surface.

(7.) In general (comp. 409, (3.)), it is easy to see that

dv
XX...S = Sror = - D-3,
do

if D denote the (scalar) semidiameter of the index surface, in the direction of de or of 7; but for the two directions of the lines of curvature, these semidiameters become (410, (3.), (4.)) the semiaxes of the index curve. Denoting then by a1 and a2 these last semiaxes, the two principal radii of curvature of any surface come by IV. to be thus expressed:

XXI... R1=aTv; R2 = a22Tv.

And if the surface be a central one, of the second order, then a1, a2 are the semiaxes of the diametral section, parallel to the tangent plane; while Tv is (comp. again 409, (3.)) the reciprocal P-1 of the perpendicular, let fall on that plane from the centre. Accordingly (comp. (6.), and 219, (4.)), it is known that the difference of the inverse squares of those semiazes varies proportionally to the product of the sines of the inclinations, of the plane of the section to the two cyclic planes.

(8.) And as regards the squares themselves, it follows from 407, LXXI., that they may be thus expressed, in terms of the principal semiaxes of the confocal surfaces, and in agreement with known results:

XXII... a1 = a2-a12; a2 = a2 - ar2;

being thus both positive for the case of an ellipsoid; both negative, for that of a double-sheeted hyperboloid; and one positive, but the other negative, for the case of an hyperboloid of one sheet (comp. 410, (15.)).

(9.) In all these cases, the normal + v is drawn towards the same side of the tangent plane, as that on which the centre o of the surface is situated (because Svp = 1); hence (by I. and XXI.) both the radii of curvature R1, R2 are drawn in this direction, or towards this side, for the ellipsoid; but one such radius for the single-sheeted hyperboloid, and both radii for the hyperboloid of two sheets, are directed towards the opposite side, as indeed is evident from the forms of these surfaces.

* Compare the second Note to page 669.

(10.) The following is another method of deducing generally the two principal curvatures of a surface, from the self-conjugate function,

XXIII... dv = φαρ, 410, IV. which affords some good practice in the processes of the present Calculus. Writing, for abridgment,

where is still a tangent to a line of curvature, the equation II. is easily brought to the form, XXV. . . - rr = ν ̄1Ѵνφτ = φτ – v-1Ѕτφν = Φr,

where I denotes a new linear and vector function, which however is not in general self-conjugate, because we have not generally φν || v. Treating then this new funetion on the plan of the Section III. ii. 6, we derive from it a new cubic equation, of the form,

XXVI... 0 M+Mr+ M"r2 + r3,

and with the coefficients,

XXVII... M = 0, Μ' = Sv1ψν, Μ" =m" – Sv1φν;

being a certain auxiliary function (= mp-1), and m" being the coefficient* analogous to M", in the cubic derived from the function & itself. The root r=0 is foreign to the present inquiry; but the two curvatures, R1, R2-1, are the two roots of the following quadratic in R ̄1, obtained from the equation XXVI. by the rejection of that foreign root:

XXVIII... 0 = (R-1Tv)2 + M"R-Tv + M.

(11.) As a first application of this general equation XXVIII., let ør have again, as in V., the form gr + Vλτμ; we shall then have the values,

and

XXIX. . . M" = 2 (g + SAUν. SμUν),

XXX... M' = (g + SAUv. SuUv)2 - (VAUv) (Vμυν),

= a great variety of transformed expressions; and the two resulting curvatures agree with those assigned by VII.

(12.) As a second application, let the surface be central of the second order, with abe for its scalar semiaxes (real or imaginary); then the symbolical cubic (350) in $ becomes,

XXXI... 0 = 43 - т" ф2 + m' - m = (p + a) ($+b2) ($ + c2); and the coefficients of the quadratic XXVIII. in R1 take the values, in which N denotes the semidiameter of the surface in the direction of the normal: XXXII.

R1+R21 - M"Tv-1 = - (m" + fUv) P= (a^2+b^2+c3-N) P;

* Compare the Note to page 673, continued in page 674. The reason of the evanescence of the coefficient M, or of the occurrence of a null root of the cubic, is that we have here Φφ ̄1ν = 0, so that the symbol -10 may represent an actual vector (comp. 351). Geometrically, this corresponds to the circumstance that when we pass, along a semidiameter prolonged, from a surface of the second order to another surface of the same kind, concentric, similar, and similarly placed, the direction of the normal does not change.

CHAP. 111.]

PRODUCT OF CURVATURES.

683

XXXIII... R1R1 = M'Tv2 = - mv ̄ = a^2b-2c-2 P4; both of which agree with known results, and admit of elementary verifications. *

(13.) In general, if we observe that m" - =x (350, XVI.), we shall see that the quadratic XXVIII. inr (or in R-Tv) may be thus written:

XXXIV... 0 = Sv-1(r2 + rxv + ψν);

or thus more briefly (comp. 398, LXXIX.),

XXXV... 0 = Sv-1(4 + r)-1 v. (14.) Accordingly, the formula XXV. gives the expression, XXXVI. . . ν2 = ($ + r) ̄1ν. Στφν;

from which, under the condition Svr = 0, the equation XXXV. follows at once. (15.) We have therefore generally, for the product of the two principal curvatures of sections of any surface at any point, the expression:

1

XXXVII. . . RR1 = 2-2 = - - =-4; which contains an important theorem of Gauss, whereto we shall presently proceed. (16.) Meanwhile we may remark that the recent analysis shows, that the squares a1, a2 (7.) of the semiaxes of the index-curve are generally the roots of the following equation,

XXXVIII. . . 0 = Sv (p + a ̄2)-1ν,

when developed as a quadratic in a3.

(17.) And that the same quadratic assigns the squares of the semiaxes of a diametral section, made by a plane - v, of the central surface of the second order which has Spop = 1 for its equation.

(18.) Accordingly, Vpop has the direction of a tangent to this surface, which is perpendicular to pat its extremity; and therefore the vector,

ΧΧΧΙΧ. . . σ =ρ-ρφρ = φρ - ρ-1 = (φ - ρ-2) ρ,

is perpendicular to the plane of the diametral section, which has the semidiameter p for a semiazis: so that it is perpendicular also to pitself. The equation,

XL... So (φ - ρ ̄2)-10 = 0,

assigns therefore the values of the squares (- p2) of the scalar semiaxes of the central section; which agrees with the formula XXXVIII.

(19.) If then a surface be derived from a given central surface of the second order, as the locus of the extremities of normals (erected at the centre) to the diametral sections of the given surface, each such normal (when real) having the length of one of the semiaxes of that section, the equation of this new surfacet (or locus) will admit of being written thus:

XLI... Sp (φ - ρ-2)-1ρ = 0.

* As an easy verification by quaternions of the expression XXXII., it may be remarked (comp. 408, (27.)), that if a, β, y be any three rectangular unit lines, then

fa+f+fy = const. = C1+C2 + c3 = a^2 + b2 + c2.

† When the given surface is an ellipsoid, this derived surface XLI. is therefore the celebrated Wave Surface of Fresnel, which will be briefly mentioned somewhat farther on.