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CHAP. III.] OSCULATING CONE, OSCULATING SPHERE.
plane to the sphere at r, and the normal plane to the curve at the same point will bisect the right line LM in the point s; we shall also have this proportion of sines,
LXI... sin LOS : sin SOM SUA: SUμp
= cos LOP: cos POM = sin PP1 : sin PP2,
if PP1, PP2 be the arcual perpendiculars from the point P of the conic on the two cyclic arcs; and this general rule for determining the position of the line os, or σ, applies even to the limiting case (27.), when that variable line becomes the axis of the osculating circle, at a minor summit of the curve.
(29.) As an example, let us suppose that the constants g, A, μ in the equation II. are connected by the relation,
-SA, whence LXIII... S (Γλρ.μρ) = 0;
the cyclic normals are therefore in this case sides of the cone, and the two planes which connect them with any third side are mutually rectangular; so that the conic is now the locus of the vertex of a right-angled spherical triangle, of which the hypotenuse is given. And by applying either the formula LXI., or the construction (28.) which it represents, we find that the trigonometric tangent of the arcual radius of the osculating small circle to such a conic, at either end of the given hypotenuse, is equal to half* the tangent of that hypotenuse itself.
(30.) It is obvious that every determination, of an osculating circle to a spherical curve, is at the same time the determination of what may be (and is) called an osculating right cone (or cone of revolution), to the cone which rests upon that curve, and has its vertex at the centre of the sphere. Applying this remark to the last example (29.), we arrive at the following theorem, which can however be otherwise deduced:
If a cone be cut in a circle by a plane perpendicular to a side, the axis of the right cone which osculates to it along that side passes through the centre of the section.
395. When a given curve of double curvature is not a spherical curve, we may propose to investigate the spheric surface which approaches to it most closely, at any assigned point. An osculating circle has been defined (389) to be the limit of a circle, which touches a given curve, or its tangent PT, at a given point P, and cuts the same curve at a near point q; while the tangent PT itself had been regarded (100) as the limit of a rectilinear secant, or as the ultimate position of the small chord PQ. It is natural then to define the osculating sphere, as being the limit of a spheric surface, which passes through the osculating circle, at a given point P of a curve, and also cuts that curve in a point Q, which is supposed to approach indefinitely to P, and ultimately to coincide with it. Accordingly we shall find that this definition conducts by quaternions to formula sufficiently sim
* This may also be inferred by limits from the formulæ (11.); in which r and a were used, provisionally, to denote a certain spherical arc and angle,
ple; and that their geometrical interpretations are consistent with known results: for example, the centre of spherical curvature, or the centre of the osculating sphere, will thus be shown to be, as usual, the point in which the polar axis (391, (5.)) touches the cusp-edge of the polar developable (391, (6.)). It will also be seen, that whereas in general, if R be a point in the normal plane (370, (8.)) to a given curve at P, we can only say that the difference of distances, RQ — RP, is small of an order higher than the first, if the chord PR be small of the first order; and whereas, even if R be on the polar axis (391, (4.)), we can only say generally that this difference of distances is small, of an order higher than the second; yet, if R be placed at the centre s of spherical curvature, the difference sQ-SP is small, of an order higher than the third: so that the distance of a near point Q, from the osculating sphere at the given point P, is generally small of the fourth order, the chord being still small of the first.
(1.) Operating with S., where A is an arbitrary line, on the vector equation 392, V. of the osculating circle, we obtain the scalar equation of a sphere through that circle under the form,
which may however, by 393, (7.), be brought to this other form, better suited to our present purpose,
e being any scalar constant, while x is still the vector of the centre K of the circle: and the vector σ of the centre s of the sphere is given by the formula,
III... σ = K+cVp"p',
which evidently expresses that this last centre is on the polar axis.
(2.) To express now that this sphere cuts the curve in a near point q, we are to substitute for the expression,
IV... w = pt = p + tp' + \t2p" + t3up", with w1 = 1;
but has been seen (in 391) to satisfy the three equations,
V... 0=Sp' (p), 0= Sp′′ (x − p) - p′2, 0=Sp ̋p' (k − p) ; reducing then, dividing by 313, and passing to the limit, we find for the osculating sphere the condition,
VI... Sp" (pk) + 3Sp'p" = cSp""p"p';
so that finally the vector σ satisfies the three scalar equations,
VII... 0 Sp (o - p), 0= $p" (o− p) - p2, 0= Sp"" (σ − p) – 3Sp′p",
by which it is completely determined, and of which the two last are seen to be the successive derivatives of the first, while that first is the equation of the normal plane :
CHAP. III.] CONSTRUCTION FOR CENTRE OF SPHERE.
whence the centre s of this sphere is (by the sub-arts. to 386, comp. 391, (6.)) the point where the polar axis кs touches the cusp-edge of the polar developable.
(3.) Differentials may be substituted for derivatives in the equations VII., which may also be thus written (comp. 391, (4.)),
VIII. . . 0 = dT (p−σ), 0=d2T(p−6), 0=d3T(p-o), if do=0 ;
the distance of a near point o of the given curve from the osculating sphere is therefore small (as above said), of an order higher than the third, if the chord PQ be small of the first order.
(4.) The two first equations VII., combined with V., give also
IX. . . 0 = Sp' (σ −k),
which express that the line Ks
0=Sp′′ (o-x), 0=$(x−p) (o −k);
perpendicular to the osculating plane and absolute
normal at P, as it ought to be, because it is part of the polar axis.
(5.) Conceiving the three points P, K, s, or their vectors ọ, k, σ, to vary together, the equations V. and VII., combined with their own derivatives, give among other results the following:
X. . . 0 =Srp = Sở p = Sở p = Sở (x - p) =Sơ”p ;
of which the geometrical interpretations are easily perceived.
(6.) Another easy combination is the following,
XI... 0 = Sk' (o+p−2k),
as appears by derivating the last equation IX., with attention to other relations; but 2-p is the vector of the extremity, say M, of the diameter of the osculating circle, drawn from the given point P: we have therefore this construction :
On the tangent KK' to the locus of the centre of the osculating circle, let fall a perpendicular from the extremity M of the diameter drawn from the given point P; this perpendicular prolonged will intersect the polar axis, in the centres of the osculating sphere to the given curve at P.
(7.) In general, the three scalar equations VII. conduct to the vector expression,
3Vp'p"Sp'p" + p2Vp"p"
XII... σ = p +
or with differentials,
XIII... σ = p+
3Vdpd2pSdpdp + dp2Vd3pdp
the scalar variable being still left arbitrary.
(8.) And if, as an example, we introduce the values for the helix,
whereof the three first occurred before, we find after some slight reductions the expression, in which a denotes again the constant inclination of the curve to the axis of the cylinder,
XV... σ = p-a'ß cosec2 a = cta - a'ß cot2 a;
but this is precisely what we found for к, in 389, VIII.; for the helix, then, the two centres, K and s, of absolute and spherical curvature, coincide.
(9.) This known result is a consequence, and may serve as an illustration, of the general construction (6.); because it is easy to infer, from what was shown in 389, (3.), respecting the locus of the centre K of the osculating circle to the helix, as being another helix on a co-axal cylinder, that the tangent KK' to this locus is perpendicular to the radius of curvature KP, while the same tangent (KK' or ') is always perpendicular (X.) to the tangent (PP' or p′) to the curve; Kк' is therefore here at right angles to the osculating plane of the given helix, or coincides with its polar axis: so that the perpendicular on it from the extremity м of the diameter of curvature falls at the point K itself, with which consequently the point s in the present case coincides, as found by calculation in (8.).
(10.) In general, if we introduce the expressions 376, VI., or the following, XVI. . . p' = s′Dsp‚_p′′= s22Ð ̧2p+s”D«p‚_p′′=s′3D‚3p + 3s ́s”D ̧3p + s ̈Ð ̧Ð,
in which & denotes the arc of the curve, but the accents still indicate derivations with respect to an arbitrary scalar t; and if we observe (comp. 380, (12.)) that the relations,
XVII... Dsp2=-1, S. D ̧pD ̧2p = 0, S. D ̧pD,3p + D ̧2p2 = 0,
in which Do2 and D,202 denote the squares of Dp and Dp, and S. DpDp denotes S(D ̧p.D ̧2p), &c., exist independently of the form of the curve; we find that s” and s" disappear from the numerator and denominator of the expression XII. for a-p, and that they have s' for a common factor: setting aside which, we have thus the simpler formulæ,
And accordingly the three scalar equations VII., which determine the centre of the osculating sphere, may now be written thus,
XIX... S(σ − p) D ̧p=0, S(a−p)D,2p + 1 = 0, $(a− p) D ̧3p = 0.
(11.) Conversely, when we have any formula involving thus the successive derivatives of the vector p taken with respect to the arc, s, we can always and easily generalize the expression, and introduce an arbitrary variable t, by inverting the equations XVI.; or by writing (comp. 390, VIII.),
XX... Dsp = s'1p', D2p=s'1(s'-'p')' = s′-2p′′ –
(12.) It may happen (comp. 379, (2.)) that the independent variable t is only proportional to s, without being equal thereto; but as we have the general relation, XXI... Di"p=s'"Ds", if s'=Dis=Tp' = const.,
it is nearly or quite as easy to effect the transformations (10.) and (11.) in the case here supposed, or to pass from t to s and reciprocally, as if we had s′ = 1.
(13.) If the vector ☛ be treated as constant in the derivations, or if we consider for a moment the centre s of the sphere as a fixed point, and attend only to the variations of distance of a point on the curve from it, then (remembering that T(p − σ)1 − (p − σ)2) we not only easily put (comp. VIII.) the three equations XIX. under the forms,
CHAP. III.] COEFFICIENT OF DEVIATION FROM SPHERE. 553
(14.) If then we write, for abridgment,
XXIV... r = T(x − p) = TD,2p-1 = radius of osculating circle;
we see that this scalar, S, must be constantly equal to unity, for every spherical curve; but that for a curve which is non-spherical, the distance sQ of a near point Q, from the centre s of the osculating sphere at P, is generally given by an expression of the form,
XXVII... SQ = R+
so that, at least for near points Q, on each side of the given point P, the curve lies without or within the sphere which osculates at that given point, according as the scalar, s, determined as above, is greater or less than unity.
(15.) In the case (12.), the formula XXVI. may be thus written,
whence, by carrying the derivations one step farther than in (8.), we find for the helix,
XXIX... S= cosec2 a > 1, or XXIX'... S-1= cot2 a > 0;
and accordingly it is easy to prove that this curve lies wholly without its osculating sphere, except at the point of osculation.
(16.) In general, the scalar S − 1, which vanishes (14.) for all spherical curves, and which enters as a coefficient into the expression XXVII. for the deviation SQ-SP of a near point of any other curve from its own osculating sphere, may be called the Coefficient of Non-Sphericity; and if or be the perpendicular from that near point q on the tangent PT to the curve at the given point P, we have then this limiting equation, by which the value of that coefficient may be expressed,
(17.) Besides the forms XVIII., other transformations of the expressions XII. XIII. for the vector σ of the centre of an osculating sphere might be assigned; but it seems sufficient here to suggest that some useful practice may be had, in proving that those expressions for a reduce themselves generally to zero, when the condition,
XXXI... Tp = const.
(18.) It may just be remarked, that as r-1 is often called (comp. 389, (4.)) the absolute curvature, or simply the curvature, of the curve in space which is considered, so R-1 is sometimes called the spherical curvature of that curve: while r and Rare called the radii* of those two curvatures respectively.
We shall soon have occasion to consider another scalar radius, which we propose to denote by the small roman letter r, of what is not uncommonly called the torsion, or the second curvature, of the same curve in space.