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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),
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),
* 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.
comp. pp. 553, 584) is
the deviation of the near point P, from the given circle (which osculates
(comp. the Note to p. 600).
(m). The curvilinear locus of the point c in (k) is one branch of
from which it follows that P is a singular point of the section here
section, with that of the developable Locus of Tangents, made by the
(n). A given Curve of double curvature is therefore generally a
(0). Additional light may be thrown on this whole theory of the
(p). Quaternion forms (pp. 603, 604) of the condition of coinci-
in which is the radius of the circle of contact, of a sphere with its
(9). The singular line into which the two branches of the curvi-
(r). With the geometrical hints thus gained from interpretation