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stant velocity, assumed = 1. And if we denote by v the point in which the given radius R or Ps is nearest to a consecutive radius of the same kind, or to the radius of a consecutive osculating sphere, then this point v divides the line Ps internally, into segments which may (ultimately) be thus expressed (pp. 580, 581),

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But these and other connected results, depending on s', have their known analogues (with H for J, and r for R), in that earlier theory (e) which introduces only s3 (besides s1 and s2): and they are all included in the general theory of emanant lines and planes (396, 397), of which some new geometrical illustrations (pp. 582-584) are here given.

(f). New auxiliary scalar n (= p-1RR' = cot J sec P= &c.), = velocity of centres of osculating sphere, if the velocity of the point P of the given curve be taken as unity (e); n vanishes with R', cot J, and (comp. 395) the coefficient S- 1 (=nrr ̄1) of non-sphericity, for the case of a spherical curve (p. 584). Arcs, first and second curvatures, and rectifying planes and lines, of the cusp-edges of the polar and rectifying* developables; these can all be expressed without going beyond s5, and some without using any higher power than s1, or differentials of the orders corresponding; r1=nr, and r1 = nr, are the scalar radii of first and second curvature of the former cusp-edge, ri being positive when that curve turns its concavity at s towards the given curve at P: determination of the point R, in which the latter cusp-edge is touched by the rectifying line A to the original curve (pp. 584-587).

(9). Equation with one arbitrary constant (p. 587), of a cone of the second order, which has its vertex at the given point P, and has contact of the third order (or four-side contact) with the cone of chords (397) from that point; equation (p. 590) of a cylinder of the second order, which has an arbitrary line PE from P as one side, and has contact of the fourth order (or five-point contact) with the curve at P; the constant above mentioned can be so determined, that the right line PE shall be a side of the cone also, and therefore a part of the intersection of cone and cylinder; and then the remaining or curvilinear part, of the complete intersection of those two surfaces of the second


The rectifying plane, of the cusp-edge of the rectifying developable, is the plane of λ and r', of which the formula LIV'. in p. 587 is the equation; and the rectifying line RH, of the same cusp-edge, intersects the absolute normal PK to the given curve, or the radius (r) of first curvature, in the point н in which that radius is nearest (e) to a consecutive radius of the same kind. But this last theorem, which is here deduced by quaternions, had been previously arrived at by M. de Saint-Venant (comp. the Note to p. xv.), through an entirely different analysis, confirmed by geometrical considerations.


order, is (by known principles) a gauche curve of the third order, or what is briefly called a Twisted Cubic: and this last curve, in virtue of its construction above described, and whatever the assumed direction of the auxiliary line PE may be, has contact of the fourth order (or five-point contact) with the given curve of double curvature at P (pp. 587-590, comp. pp. 563, 572).

(h). Determination (p. 590) of the constant in the equation of the cone (g), so that this cone may have contact of the fourth order (or fire-side contact) with the cone of chords from P; the cone thus found may be called the Osculating Oblique Cone (comp. 397), of the second order, to that cone of chords; and the coefficients of its equation involve only r, I, r', r′, r", r′′, but not r", although this last derivative is of no higher order than r", since each depends only on 85 (and lower powers), or introduces only fifth differentials. Again, the cylinder (g) will have contact of the fifth order (or six-point contact) with the given curve at P, if the line PE, which is by construction a side of that cylinder, and has hitherto had an arbitrary direction, be now obliged to be a side of a certain cubic cone, of which the equation (p. 590) involves as constants not only rrr'r'r'r", like that of the osculating cone just determined, but also r". The two cones last mentioned have the tangent (7) to the given curve for a common side,† but they have also three other common sides, whereof one at least is real, since they are assigned by a cubic equation (same p. 590); and by taking this side for the line PE in (g), there results a new cylinder of the second order, which cuts the osculating oblique cone, partly in that right line PE itself, and partly in a gauche curve of the third order, which it is proposed to call an Osculating Twisted Cubic (comp. again (g)), because it has contact of the fifth order (or six-point contact) with the given curve at P (pp. 590, 591).

(i). In general, and independently of any question of osculation, a Twisted Cubic (g), if passing through the origin o, may be represented by any one of the vector equations (pp. 592, 593),

xix Pages.

* By Dr. Salmon, in his excellent Treatise on Analytic Geometry of Three Dimensions (Dublin, 1862), which is several times cited in the Notes to this final Chapter (III. iii.) of these Elements. The gauche curves, above mentioned, have been studied with much success, of late years, by M. Chasles, Sig. Cremona, and other geometers: but their existence, and some of their leading properties, appear to have been first perceived and published by Prof. Möbius (see his Barycentric Calculus, Leipzig, 1827, pp. 114–122, especially p. 117).

This side, however, counts as three (p. 614), in the system of the six lines of intersection (real or imaginary) of these two cones, which have a common vertex P, and are respectively of the second and third orders (or degrees). Additional light will be thrown on this whole subject, in the following Series (399); in which also it will be shown that there is only one osculating twisted cubic, at a given point, to a given curve of double curvature; and that this cubic curve can be determined, without resolving any cubic or other equation.

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it is ultimately equal (p. 595) to the quarter of the deviation (397)

of the same near point P, from the osculating circle at P, multiplied by

the sine of the small angle spss, which the small are ss, of the locus of

the spheric centre s (or of the cusp-edge of the polar developable) sub-

tends at the same point P; and it has an outward or an inward direc-

tion, according as this last arc is concave or convex (ƒ) ats, towards the

given curve at P (pp. 585, 595). It is also ultimately equal (p. 596)

to the deviation PS, PS, of the given point P from the near sphere,

which osculates at the near point Ps; and likewise (p. 597) to the com-

ponent, in the direction of SP, of the deviation of that near point from

the osculating circle at P, measured in a direction parallel to the nor-

mal plane at that point, if this last deviation be now expressed to the

accuracy of the fourth order: whereas it has hitherto been considered

sufficient to develope this deviation from the osculating circle (397) as

far as the third order (or third dimension of s); and therefore to treat

it as having a direction, tangential to the osculating sphere (comp.

pp. 566, 594).

(k). The deviation (A1) is also equal to the third part (p. 598) of

the deviation of the near point P, from the given circle (which osculates
at P), if measured in the near normal plane (at Ps), and decomposed in
the direction of the radius R, of the near sphere; or to the third part
(with direction preserved) of the deviation of the new near point in
which the given circle is cut by the near plane, from the near sphere: or
finally to the third part (as before, and still with an unchanged direc-

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from which it follows that P is a singular point of the section here
considered, but not a cusp of that section, although the curvature
at P is infinite: the ordinate DC varying ultimately as the power
with exponent of the abscissa PD. Contrast (pp. 600, 601), of this


section, with that of the developable Locus of Tangents, made by the
same normal plane at P to the given curve; the vectors analogous to
PD and DC are in this case nearly equal to 82 and -3r1v; so
that the latter varies ultimately as the power of the former, and the
point P is (as it is known to be) a cusp of this last section.

(n). A given Curve of double curvature is therefore generally a
Singular Line (p. 601), although not a cusp-edge, upon that Surface (1),
which is at once the Locus of its osculating Circle, and the Envelope
of its osculating Sphere: and the new developable surface (d), as being
circumscribed to this superficial locus (or envelope), so as to touch it
along this singular line (p. 612), may naturally be called, as above,
the Circumscribed Developable (p. 581).

(0). Additional light may be thrown on this whole theory of the
singular line (n), by considering (pp. 601-611) a problem which was
discussed by Monge, in two distinct Sections (xxii. xxvi.) of his well-
known Analyse (comp. the Notes to pp. 602, 603, 609, 610 of these
Elements); namely, to determine the envelope of a sphere with varying
radius R, whereof the centre s traverses a given curve in space; or
briefly, to find the Envelope of a Sphere with One varying Parameter
(comp. p. 624): especially for the Case of Coincidence (p. 603, &c.), of
what are usually two distinct branches (p. 602) of a certain Charac-
teristic Curve (or arête de rebroussement), namely the curvilinear enve-
lope (real or imaginary) of all the circles, along which the superficial
envelope of the spheres is touched by those spheres themselves.

(p). Quaternion forms (pp. 603, 604) of the condition of coinci-
dence (o); one of these can be at once translated into Monge's equa-
tion of condition (p. 603), or into an equation slightly more general,
as leaving the independent variable arbitrary; but a simpler and
more easily interpretable form is the following (p. 604),

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in which is the radius of the circle of contact, of a sphere with its
envelope (0), while r1 is the radius of (first) curvature of the curve (s),
which is the locus of the centre s of the sphere.

(9). The singular line into which the two branches of the curvi-
linear envelope are fused, when this condition is satisfied, is in general
an orthogonal trajectory (p. 607) to the osculating planes of the curve
(s); that curve, which is now the given one, is therefore (comp. 391,
395) the cusp-edge (p. 607) of the polar developable, corresponding to
the singular line just mentioned, or to what may be called the curve
(P), which was formerly the given curve. In this way there arise
many verifications of formula (pp. 607, 608); for example, the
equation (G1) is easily shown to be consistent with the results of (ƒ).

(r). With the geometrical hints thus gained from interpretation
of quaternion results, there is now no difficulty in assigning the Com-
plete and General Integral of the Equation of Condition (p), which was
presented by Monge under the form (comp. p. 603) of a non-linear
differential equation of the second order, involving three variables


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