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CHAP. III. VECTOR EQUATION OF SURFACE OF CENTRES. 689
as follows, using a new expression for ø, in terms of or of p, which may then be transformed into a function of two independent and scalar variables. Denoting (comp. (32.)) by ai, bi, ci the semiaxes of the confocal which cuts the given surface in the given line of curvature, and by a2, b2, c2 those of the other confocal, so that the normals v1, v2 to these two confocals have the directions of the tangents r', lately considered, we have not only the expressions LXXXI. for r'-1, with a'b'c' changed to a, b, c, but also the analogous expressions (comp. 407, LXXI.),
LXXXVIII... r ́1 = a2 — a22 = b2 — b22 = c2 — c22.
We have therefore by XLII., combined with 407, XVI., this very simple expression for σ: ·(Ø ̄1 + r−1) = $2 ̄1v = $2 ̈1¢p ;
containing, in the present notation, and as a result of the present analysis, a known and interesting theorem,* on which however we cannot here delay.
(39.) It follows from this last value of o, combined with the expression 408, LXXXII. for p, that we may write,
as the sought Vector Equation of the Surface of Centres of curvature of a given quadric (abe); ambiguous signs being virtually included in these three terms, because in the subsequent eliminations† the semiaxes enter only by their squares: while l, a, a' are constants, as in 407, &c., for the whole confocal system, and abc are also constant here, but a2 - a12 and a2 — a1⁄22, or r′-1 and r-1 (38.), are variable, and may be considered to be the two independent scalars of which σ is a vector func
413. Some brief remarks may here be made, on the connexion of the general formula,
in which r=RTv (412, XXIV.), and which when developed by the rules of the Section III. ii. 6 takes (comp. 398, LXXIX.) the form of the quadratic,
* Namely Dr. Salmon's theorem (page 161 of his Treatise), that the centres of curvature of a given quadric at a given point are the poles of the tangent plane, with respect to the two confocals. The connected theorem (page 136), respecting the rectilinear locus of the poles of a given plane, with respect to the surfaces of a confocal system, is at once deducible from the quaternion expression 407, XVI. for ply, although the theorem did not happen to be known to the present writer, or at least remembered by him, when he investigated that formula of inversion for other applications, of which some have been already given.
+ The corresponding elimination in co-ordinates was first effected by Dr. Salmon, who thus determined the equation of the surface of centres of curvature of a quadric to be one of the twelfth degree. (Compare pages 161, 162 of his already cited Treatise.)
II... 2+rSv1xv + Sv ̄1yv = 0, 412, XXXIV. with Gauss's theory of the Measure of Curvature of a Surface; and especially with his fundamental result, that this measure is equal to the product of the two principal curvatures of sections of that surface: a relation which, in our notations, may be thus expressed,
III. . . V.dU» ¿ ̄v = R11R2¬1Vdpèp,
(1.) As regards the deduction, by quaternions, of the equation III., in which d and may be regarded as two† distinct symbols of differentiation, performed with respect to two independent scalar variables, we may observe that, by principles and rules already established,
and that therefore the first member of III. may be thus transformed:
(2.) Again, since we have dy = pdp (410, IV., &c.), and in like manner &v= øp, the relations Svdp = 0, Svdp = 0, and the self-conjugate property of p, allow us to write,
VI... Vdvdv = Vdoop, and VII... Vdpop v ̈1Svdpèp ; whence follows at once by V. the formula III., if we remember the general expression, deduced from the quadratic II.,
(3.) If then we suppose that P, P1, P2 are any three near points on an arbitrary surface, and that R, R1, R2 are three near and corresponding points on the unit sphere, determined by the condition of parallelism of the radii OR, OR1, org to the normals PN, PINI, P2N2, the two small triangles thus formed will bear to each other the ultimate ratio,
a result which justifies (although by an entirely new analysis) the adoption by Gauss
The reader is referred to the Additions to Liouville's Monge (pages 505, &c.), in which the beautiful Memoir by Gauss, entitled: Disquisitiones generales circa superficies curvas, is with great good taste reprinted in the Latin, from the Commentationes recentiores of the Royal Society of Göttingen. He is also supposed to look back, if necessary, to the Section III. ii. 6 of these Elements (pages 435, &c.), and especially to the deduction in page 437 of from ø, remembering that the latter function (and therefore also the former) is here self-conjugate.
+ Compare page 487, and the Note to page 684.
CHAP. III.] MEASURE OF CURvature of a surface.
of this product* of curvatures of sections, as the measure of the curvature of the surface, with his signification of the phrase.
(4.) As another form of this important product or measure, if we conceive that the vector p of the surface is expressed as a function (372) of two independent scalars, t and u, and if we write for abridgment,
X... Dtp = p', Dup = P1, Di2p=p", D¿Dup=p,, Du2p=Pu
which will allow us (comp. 372, V.) to assume for the normal vector the expression,
rt - 82
XIV. . . p = ix + jy + kz, p' =Dxp=i+kp, p,= Dyp=j+kq;
(5.) In general, the equation XII. may be thus transformed,
XVI. . . v1R1 ̄1R2 ̄1 = S (Vvp". V vp,) – (Vvp,')2 + v2 (Sp'p., -p,'2);
if XVIII...e=-p'2, ƒ=-Sp′p,, g=-p2, whence XIX... v2 = ƒ2 — eg and if we still denote, as in X., derivations relatively to t and u by upper and lower accents, we may substitute in the quadruple of the equation XVI. the values,
XX. . . 2Vvp” = (-2f) tiếp, 2Vvp=-gp tep, 2V
XXI. . . 2 (Sp”p„ - p‚2) = e„ - 2ƒ, +9" ;
=-gp +(2f-9') P
hence the measure of curvature is an explicit function of the ten scalars,
XXII. e, f, 9 ; é, ƒ“', 9' ; è̟f9,; and e„-2ƒ,'+9′′:
and therefore, as was otherwise proved by Gauss, this measure depends only‡ on the
* If it be supposed to be in any manner known that a limit such as IX. exists, or that the quotient of the two vector areas in III. is a scalar independent of the directions of PP1, PP2, or of dp, dp, we have only to assume that these are the directions of the lines of curvature, in order to obtain at once, by 412, II., the product R1-R2 as the value of this quotient or limit.
The quadratic in R-1 may be formed by operating on 412, II. with S.p' and S.p, and then eliminating dt: du.
The proof by quaternions, above given, of this exclusive dependence, is perhaps as simple as the subject will allow, and is somewhat shorter than the corresponding proof in the Lectures: in page 605 of which is given however the equation,
expression (XVII.) of the square of a linear element, în terms of two independent scalars (t, x), and of their differentials (dt, du).
(6.) Hence follow also these two other theorems* of Gauss:—
If a surface be considered as an infinitely thin solid, and supposed to be flexible but inextensible, then every deformation of it, as such, will leave unaltered, Ist, the Measure of Curvature at any Point, and IInd, the Total Curvature of any Area ; that is, the area of the corresponding portion of the unit sphere, determined as in (3.) by radii parallel to normals.
(7.) Supposing now that t and u are geodetic co-ordinates, whereof the former represents the length of a geodetic AP from a fixed point A of the surface, and the latter represents the angle BAP which this variable geodetic makes at a with a fized geodetic AB, it is easy to see that the general expression XVII. takes the shorter form,
XXIII. . . Tdp2 = dt2 + n2du2, in which XXIV... n = Tp, = Tv ;
so that we have now the values,
XXV... e = 1, f=0, g=n2, g'= 2nn', g'' = 2nn" + 2n′2,
and the derivatives of e and fall vanish.
And thus the general expression XII. for the measure of curvature reduces itself by (5.) to the very simple form,
in which n is generally a function of both t and u, although here twice derivated with respect to the former only.
(8.) The point r being denoted by the symbol (t, u), and any other point p' of the surface by (t + At, u + Au), we may consider the two connected points P1, P2, of which the corresponding symbols are (t + At, u) and (t, u+▲u); and then the quadrilateral PPP'P2, bounded by two portions PP1, P2P′ of geodetic lines from a, and (as we may suppose) by two arcs PP2, PIP' of geodetic circles round the same fixed point, will have its area ultimately = nAtAu (by XXIII.), and therefore (by XXVI., comp. (3.), (6.)) its total curvature ultimately - n"AtAu, or =— Am'.Av, when At and Au diminish together, by an approach of P' to P.
(9.) Again, in the immediate neighbourhood of A, we have n=t, n'= 1; changing then-An' to din', and integrating with respect to t from t=0, we obtain 1-n' as the coefficient of Au in the result, and are thus conducted to the expression:
XXVII... Total Curvature of Triangle APP' = (1 − n') Au, ultimately,
if AP, AP' be any two geodetic lines, making with each other a small angle = Au, and if PP' be any small arc (geodetic or not) on the same surface.
4 (eg-f2)2R1'R2-1 = e (g22 — 29,ƒ" +g,e,)
+g(e,2-2e'ƒ, +é'g') - 2 (eg -ƒ2) (e,,- 2ƒ,' +9′′),
which may now be deduced at sight from XVI., by the substitutions XIX. XX. XXI., and differs only in notation from the equation of Gauss (Liouville's Monge, page 523, or Salmon, page 309).
CHAP. III.] TOTAal curvature, sPHEROIDAL EXCESS.
(10.) Conceive then that PQ is a finite arc of any curve upon the surface, for which therefore t, and consequently n', may be conceived to be a function of u; we shall have this other expression of the same kind,
XXVIII... Total Curvature of Area APQ= [(1 − n') du = ▲u — f n'du ;
the area here considered being bounded by the two geodetic lines AP, AQ, which make with each other the finite angle Au, and by the arc PQ of the arbitrary curve. (11.) If this curve be itself a geodetic, and if we treat its co-ordinates t, u, and its vector p, as functions of its arc, s, then the second differential of p, namely, XXIX. . . d2p = p'd2t + p ̧d2u + p′′dt2 + 2p,'dtdu+ p„„du2,
must be normal to the surface at P, and consequently perpendicular to p' and p Operating* therefore with S.p', and attending to the relations XVIII. and XXV., which give
XXX... p22 = − 1, Sp'p, = Sp'p" = Sp'p', = 0, Sp'p„,= — Sp,p,' = nn', we obtain the differential equation,
XXXI. . . d2t = nn'du2, or XXXII... dv: - n'du,
if we observe that we may write,
XXXIII... dt = cos vds, ndusin vds, because XXXIV. . . dt2 + n2du2 = ds? ; v being here the variable angle, which the geodetic PQ makes at P with AP prolonged.
(12.) Substituting then for n'du, in XXVIII., its value de given by XXXII., the integration becomes possible, and the result is Au+ Av; where Au is still the angle at A, and π + Av = (π − v) + (v + Av) is the sum of the angles at P and Q, in the geodetic triangle APQ.
(13.) Writing then B and C instead of P and Q, we thus arrive at another most remarkable Theorem† of Gauss, which may be expressed by the formula:
XXXV... Total Curvature of a Geodetic Triangle ABC = A + B + C − π,
what may be called the Spheroidal Excess; A, B, C, in the second member, being used to denote the three angles of the triangle: and the total surface of the unit sphere (=47) being represented by 720°, when the part corresponding to the geodetic triangle is thus represented by the angular excess, A + B + C − 180°.
(14.) And it is easy to perceive, on the one hand, how this theorem admits of being extended, as it was by Gauss, to all geodetic polygons: and on the other hand, how it may require to be modified, as it was by the same eminent geometer, so as to give what would on the same plan be called a spheroidal defect, when the measure of curvature is negative, as it is for surfaces (or parts of surfaces) of which the principal sections have their curvatures oppositely directed.
*To operate with S.p, would give a result not quite so simple, but reducible to the form XXXI., with the help of d2s = 0.
The enunciation of this theorem, respecting which its illustrious discoverer justly says, "Hoc theorema, quod, ni fallimur, ad elegantissima in theoria superficierum curvarum referendum esse videtur,"... is given in page 533 of the Additions to