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414. The only sections of a surface, of which the curvatures have been above determined, are the two principal normal sections at any proposed point; but the general expressions of III. iii. 6 may be applied to find the curvature of any plane section, normal or oblique, and therefore also of any curve on a given surface, when only its osculating plane is known. Denoting (as in 389, &c.) by p and the vectors of the given point P, and of the centre K of the osculating circle at that point, and by s the arc of the curve, we have generally (by 389, XII. and VI.),

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I... Vector of Curvature of Curve = ÅP ́1 = (p − k) ́1 = D, p =

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the independent variable in the last expression being arbitrary. And if we denote by ☛ and § the vectors of the points s and x, in which the axis of the osculating circle meets respectively the normal and the tangent plane to the given surface, we shall have also, by the right-angled triangles, the general decomposition, KP SP+XP(as vectors), or

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where the two components admit of being transformed as follows: III... Normal Component of Vector of Curvature of Curve (or

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= Vector of Normal Curvature of Surface for the direction of

the given tangent;

1, being the vectors of the centres 81, 8, (comp. 412) of the two principal curvatures, and v being the angle at which the curve (or its tangent dp) crosses the first line of curvature (or its tangent 71), while is the vector of the centre s of the sphere which is said to osculate to the surface, in the given direction (of dp); and

IV... Tangential Component of Vector of Curvature
= (p)1 = v1dp 'Svdp1d2p

=Vector of Geodetic Curvature of Curve (or Section);

this latter vector being here so called, because in fact its tensor re

Liouville's Monge. A proof by quaternions was published in the Lectures (pages 606-609, see also the few preceding pages), but the writer conceives that the one given above will be found to be not only shorter, but more clear.



presents what is known by the name of the geodetic* curvature of a curve upon a surface: the independent variable being still arbitrary.

(1.) As regards the decomposition II., if a, ß be any two rectangular vectors OA, OB, and if y = oc = the perpendicular from o on AB, then (comp. 316, L., and 408, XLI.),

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(2.) To prove the first transformation III., we have, by I. and II., observing that dSvdp = 0,

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(3.) Hence, by 412, (7.), if we denote the vector III. of normal curvature by R1U, we have the general expressions (comp. 412, I. XXI.),

VII... σ = p + RUv, _R=D2.Tv, with VIII. . . Tv = P ́1,

for the case of a central quadric; D being generally the semidiameter of the index surface (410, (9.), &c.), or for a quadric the semidiameter of that surface itself, which has the direction of the tangent (or of dp): and P being, for the latter surface, the perpendicular from the centre on the tangent plane, as in some earlier formulæ.

(4.) To deduce the second transformation III., which contains a theorem of Euler, let 7, T1, T2 denote unit tangents to the section and the two lines of curvature, so that

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XI... ST1971 = R1 ̄1Tv, ST2$T2=R2 ̄1Tv, ST1&T2= ST2PT1 = 0;

XII... R R1 ̄1 cos2 v + R21 sin2 v,


and the required transformation is accomplished.

(5.) The theorem of Meusnier may be considered to be a result of the elimination (2.) of d2p from the expressions for the normal component III. of what we may call the Vector Dap of Oblique Curvature; and it may be expressed by the equation,

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if it be now understood that the point s, of which σ is the vector, is the centre of the

* The name, "courbure géodésique," was introduced by M. Liouville, and has been adopted by several other mathematical writers. Compare pages 568, 575, &c. of his Additions to Monge.

circle which osculates to the normal section; or of the sphere which osculates in the same direction to the surface, as will be more clearly seen by what follows. (6.) In general, if p+ Ap be the vector of any second point P′ of the given surface, the equation

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XIV... S
- S with a for a variable vector,
w-p Δρ

represents rigorously the sphere which touches the surface at the given point P, and passes through the second point P'; conceiving then that the latter point approaches to the former, and observing that the development* by Taylor's Series of the equation fp const. gives (if dfp = 2Svdp, and dv=ødp),


XV... 0 = Ap 2Afp = 28

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+ 8940


+ terms which vanish generally with Ap, Δρ

even if they be not always null, we are conducted in a new way, by the known conception of the Osculating Sphere for a given direction to a surface, to the same centre s, and radius R, as before: the equation of this sphere being,

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(7.) Conversely, if we assume a radius R, such that R1 is algebraically intermediate between R1 and R2, the tangent sphere,

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will cut the surface in two directions of osculation, assigned by the formula XII.; but if R-1 be outside those limits, there will be only contact, and not any (real) intersection, at least in the vicinity of P.

(8.) If p' be again, as in (6.), any second point of the surface, and if we denote for a moment by (II) and (2) the normal plane PNP' and the normal section corresponding, we may suppose that N is the point in which the normals to the plane curve (2) at P and P' intersect; and if we then erect a perpendicular at N to the plane (II), it will be crossed by every perpendicular at P' to the tangent P'T' to the section, and therefore in particular by the normal at P' to the surface, in a point which we may call N': so that the line P'N is the projection, on the plane PP'N, of this second normal P'N' to the surface. Conceiving then the plane (II) to be fixed, but the point r' to approach indefinitely to P, we see that the centre s of curvature of the normal section (2), which is also by (6.) the centre of the osculating sphere to the surface for the same direction, is the limiting position of the point x, in which

*Compare Art. 374, and the Second Note to page 508. The occasional use, there mentioned, of the differential symbol dp as signifying a finite and chordal vector, in the development of ƒ(p +dp), has appeared obscure, in the Lectures, to some friends of the writer; and he has therefore aimed, for the sake of clearness, in at least the text of these Elements, and especially in the geometrical applications, to confine that symbol to its first signification (100, 369, 373, &c.), as denoting a tangential vector (finite or infinitely small, and to a curve or surface): p itself being generally regarded as a vector function, and not as an independent variable (comp. 362, (3.)).


the given normal at P is intersected by the projection* of the near normal P'N', on the given normal plane.

(9.) The two components III. and IV. are included in the binomial expression, XVIII... Vector of Oblique Curvature (or of Curvature of Oblique Section) = (p − k) ̄1 = v−1Sdvdp‍1 + v ̄1dp ̄1Svdp ̄1d2p,

which is obtained by substituting in I. the general equivalent 409, XXI. for d2p, and in which (as before) the independent variable is arbitrary; and the tangential component IV. may be otherwise found by observing that, by I. and II.,

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(10.) Another way of deducing the same component IV., is to resolve the following system of three scalar equations, which by the geometrical definition of the point x the vector must satisfy :

XX... S(3-p)v=0; $(% − p)dp=0; $(%-p)d2p = dp2;

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or (p −)-1= &c., as before. We have also the transformations,

XXII.. Vector of Geodetic Curvature = (p − §)-1

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(11.) The definition of the point x shows also easily, that if a developable surface (D) be circumscribed to a given surface (s), along a given curve (c), and if, in the unfolding of the former surface, the point x be carried with the tangent plane, originally drawn to the latter surface at P, it will become the centre of curvature, at the new point (P), to the new or plane curve (c') obtained by this development: so that the radius (PX) of geodetic curvature is equal, as indeed it is known to be, to the radius of plane curvature of the developed curve.

(12.) This plane curve (c') is therefore a circle‡ (or part of one) if the condition,

XXIII... PX = T (− p) = const.,

* The reader may compare the calculations and constructions, in pages 600, 601 of the Lectures. In the language of infinitesimals, an infinitely near normal P'N' intersects the axis of the osculating circle, to the given normal section.

† Compare page 576 of the Additions to Liouville's Monge.

The curves on any given surface, which thus become circles by development, have also the isoperimetrical property expressed in quaternions (comp. the first Note to page 530) by the formula,

XXVI... [S(Uv.dpôp)+cô [ Tập = 0,

which conducts to the differential equation,

XXVII... c-ldp = V.Uv dUdp (comp. 380, IV.),

be satisfied; but it degenerates into a right line, if this radius of geodetic curvature be infinite, that is, if

XXIV... T(p-2)-1=0, or XXV. . . Svdpd2p = 0,

or finally (by 380, II., comp. 409, XXV.), if the original curve (c) be a geodetic line on the given surface (s), and therefore also on the developable (D): which agrees with the fundamental property (382, 383) of geodetics on a developable surface. (13.) Accordingly it may be here observed that the general formula IV., combined with the notations and calculations of 382, conducts to the expression zdx + dv (z+v) Tp', or for the geodetic curvature of any curve on a developable ds

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surface, whereof the element ds crosses a generating line at the variable angle ₺, while zdx is the angle between two such consecutive lines: a result easily confirmed by geometrical considerations, and agreeing with the differential equation z + v'=0 (382, IX.) of geodetics on a developable.

415. We shall conclude the present Section with a few supplementary remarks, including a new and simplified proof of an important theorem (354), which we have had frequent occasion to employ for purposes of geometry, and which presents itself often in physical applications of quaternions also: namely, that if the linear and vector function be self-conjugate, then the Vector Quadratic,

I... Vpop = 0,

354, I. represents generally a System of Three Real and Rectangular Directions; and that these (comp. 405, (1.), (2.), &c.) are the directions of the Axes of the Central Surfaces of the Second Order, which are represented by the scalar equation,

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III... Spøp = Cp2+ C', where C and C' are any two scalar constants.

(1.) It is an easy consequence of the theory (350) of the symbolic and cubie equation in 4, that if c be a root of the derived algebraical cubic M=0 (354), and if we write = + c (as in that Article), the new linear and vector function p must be reducible to the binomial form (351),

and in which the scalar constant c can be shown to have the value,


XXVIII.. .. c = ( − p) U.vdp = + T (§ − p) = Radius of Geodetic Curvature,

radius of developed circle; and each such curve includes, by XXVI., on the given surface, a maximum area with a given perimeter on which account, and in allusion to a well-known classical story, the writer ventured to propose, in page 582 of the Lectures, the name "Didonia" for a curve of this kind, while acknowledging that the curves themselves had been discovered and discussed by M. Delaunay.

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