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CHAP. III. VECTOR EQUATION OF OSCULATING circle.
=x+y− }(x8+yt) p′′p'-1 + &c. :
in which last equation, both members are generally quaternions.
(5.) The comparison of the scalar parts gives here no useful information, on account of the arbitrary character of the coefficients x and y; but these disappear, with the two other scalars, s and t, in the comparison of the vector parts, whence follows the determinate and limiting equation,
XIII... 2Vp' (w− p)`1 = — Vp′′p' ̄",
which evidently agrees with V.
(6.) It is then found, by this little quaternion calculation, as was of course to be expected, that the circle (3.), through any three near points of a curve in space, coincides ultimately with the osculating circle, if the latter be still defined (389) with reference to a given tangent, and a near point, which tends to coincide with the given point of contact.
393. An osculating circle to a curve of double curvature does not generally meet that curve again; but it intersects generally a plane curve, of the degree n, to which it osculates, in 2n − 3 points, distinct from the point P of osculation, whereof one at least must be real, although it may happen to coincide with that point P: and such a circle intersects also generally a spherical curve of double curvature, and of the degree n, in n-3 other points, namely in those where the osculating plane to the curve meets it again. An example of each of these two last cases, as treated by quaternions, may be useful.
(1.) In general, if we clear the recent equation, 392, V. or XIII., of fractions, it becomes,
I... 0 = 2p′2Vp' (w−p) + (w−p)2Vp′′p';
in which p = OP = the vector of the given point of osculation, and p', p" are its first and second derivatives, taken with respect to any scalar variable t, and for the particular value (whether zero or not) of that variable, which answers to the particular point P; while o denotes generally the vector of any point upon the circle, which osculates to the given curve at that point P.
(2.) Writing then (comp. 389, (10.)),
II. . . p = ta + t1ß, p' = a - t2ß, p" = 2t3ß,
to express that we are seeking for the remaining intersection Q of a plane hyperbola
* This conclusion is indeed so well known, and follows so obviously from the doctrine of infinitesimals, that it is only deduced here as a verification of previous formulæ, and for the sake of practice in the present Calculus.
with its osculating circle at P, the equation I. becomes, after a few easy reductions, including a division by Vaß, the following biquadratic in x,
in which the cubic factor is to be set aside, as answering only to the point P itself. (3.) Substituting then, in III., the remaining value IV. of x, we find the expresssion,
comparing which with 371, (3.), we see that if the tangent to the hyperbola at the given point r intersects the asymptotes in the points A, B, then the tangent at the sought point q meets the same lines OA, OB in points A', B', such that
VI... OA. OA' = OB2, OB. OL' = OA";
whence q is at once found, as the bisecting point of the line A'B'.
(4.) A still more simple construction, and one more obviously agreeing with known results, may be derived from the following expression for the chord rQ:
VII... PQ = w - p = (t23ß ̃ ̄2 — t ̄2a ̄2) (ta2ß — t11aß2)
whence it follows (comp. 226) that if this chord PQ, both ways prolonged, meets the two asymptotes OB and OA in the points R and 8, we have then the inverse similitude of triangles (118),
VIII... A ROS 'AOB.
(5.) As regards the equality of the intercepts, RP and Qs, it can be verified without specifying the second point q on the hyperbola, or the second scalar, æ, by observing that the formula III., combined with the first equation II., conducts to the expressions,
(6.) And as regards the general reduction, of the determination of the osculating circle to a spherical curve of double curvature, to the determination of the osculating plane, it is sufficient to observe that when we take the centre of the sphere for the origin, and therefore write (comp. 381, XIV.),
XI... p2 = const., Spp' = 0, Spp"-p",
then if we operate on the vector equation I. with the symbol V. - p3, there results the scalar equation,
XII... 0=2Sp(w− p) + (w − p)2 = w2 — p2,
and divide by
which expresses that the circle is entirely contained on the same spheric* surface as the curve; while the other scalar equation,
XIII. . . . 0 = Sp"p'(w− p),
obtained by operating on I. with S. p", expresses (comp. 376, V.) that the same
* This conclusion is geometrically evident, but is here drawn as above, for the sake of practice in the quaternions.
CASE OF A SPHERICAL CURVE.
circle is in the osculating plane:* so that its centre K is the foot of the perpendicular let fall on that plane from the origin, and we may therefore write (comp. 385, VI.),
XIV... OK = K = →→→
and with the verification that the expression XIV. agrees with the general formula, 389, IV.,
XVI. . . pVp"p' + p′3 = Sp′′p'p,
when the conditions XI. are satisfied.
(7.) And even if the given curve be not a spherical one, yet if we retain the general expression for k,
and operate on I. with S.p" and S. p′′p', we find again the equation XIII. of the osculating plane, combined with a new scalar equation, which may after a few reductions be written thus,
XVIII... (k)2 = (p − k)2 ;
and which represents a new sphere, whereon the osculating circle to the curve is a great circle.
394. To give now an example of a spherical curve of double curvature, with its osculating circle and plane for any proposed point P, and with a determination of the point q in which these meet the curve again (393), we may consider that spherical conic, or spheroconic, of which the equations are (comp. 357, II.),
namely the intersection of the sphere, which has its centre at the origin, and its radius =r, with a cone of the second order, which has the same origin for vertex, and has the given lines A and μ for its two (real) cyclic normals. And thus we shall be led to some sufficiently simple spherical constructions, which include, as their plane limits, the analogous constructions recently assigned for the case of the common hyperbola.
(1.) Since Sλpup = 2SλpSμp — p2Sλμ (comp. 357, II'.), the equations I. and II. allow us to write, as their first derivatives, or at least as equations consistent therewith,
III... Spp'= 0, Sλp' + Sλp = 0, Sμр' - Sμp=0,
because the independent variable is here arbitrary, so that we may conceive the first derived vector p' to be multiplied by any convenient scalar; in fact, it is only the
* Compare the Note immediately preceding.
direction of this tangential vector p' which is here important, although we must continue the derivations consistently, and so must write, as consequences of III., the equations,
IV... Spp" + p2 = 0, Sλp"+Sλp'=0, Sup" - Sup'= 0.
(2.) Introducing then the auxiliary vectors,
V...n=V\μ‚_o=\Sμp+μ$\p, ==p+p', v=p-p';
VI... 0 = Snσ = Sλr = Sμv, Spo=2SXpSμp, Sμr=2Sμp, Sλv=2SAP, T2=v2 = p2+p",
and by new derivations,
VII... σ'=Vnp, t'=p'+p", v'=p'-p", Sλr'=Sμv' = 0,
Sλv' = - Sλv,
we see first that r and v are the vectors or and ou of the points in which the rectilinear tangent to the curve at P meets the two cyclic planes, perpendicular respectively to λ and μ; and because the radius or is seen to be the perpendicular bisector of the linear intercept TU between those two planes, so that
if the tangent arc on the sphere, to the same conic at the same point P, meet the two cyclic arcs CA and CB in the points A and B: the intercepted arc AB being thus bisected at its point of contact P, which is a well-known property of such a curve.
(3.) Another known property of a sphero-conic is, that for any one such curve the sum of the two spherical angles CAB and ABC, and therefore also the area of the spherical triangle ABC, is constant. We can only here remark, in passing, that quaternions recognise this property, under the form (comp. II.),
XI... cos (A + B) = − SUλрup = −g: Tλμ = const.
(4.) The scalar equations III. and IV. give immediately the vector expressions, (02 + p22) Vλμ Όλμο
or by (2.),
Vp (ASμp +μS\p)
Προ XIV... p = Sup'
XIII. p" = p
the new auxiliary vector being thus that of the point x, in which the osculating plane to the conic at P meets the line ŋ of intersection of the cyclic planes: so that we have the geometrical expressions,
and the lines* and v′ are the traces of the osculating plane on those two cyclic
*We may also consider the derived vectors r' and v', or the lines XT and XU, as corresponding tangents, at the points T and u (2.), to the two sections, made by the cyclic planes, of that developable surface which is the locus of the tangents TPU to the spherical conic in question.
CHAP. III.] CONSTRUCTION FOR A SPHERICAL CONIC.
planes, or of the latter on the former; while σ and σ', as being perpendicular respectively to p' and p, while each↓n, are the traces on the plane Aμ of the two cyclic normals, of the normal plane to the conic at the point P, and of the tangent plane to the sphere at that point: or at least these lines have the directions of those
(5.) Already, from the expression XVI. for the portion ox of the radius oc (2.), or of that radius prolonged, which is cut off by the osculating plane at P, we can derive a simple construction for the position of the spherical centre, or pole, say E, of the small circle which osculates at that point P, to the proposed sphero-conic. For if we take the radius r for unity, we have the trigonometric expressions, XVIII... sec CE COS EP= (T = Tr2: SUn-1p =) sec2 PB sec CP;
or letting fall (comp. Fig. 80) the perpendicular CD on the normal arc PE,
XIX... cOS DE COS DP COS PB. COS PB COS PECOS DB COS BE;
(6.) But although it is a perfectly legitimate process to mix thus spherical trigonometry with quaternions (since in fact the latter include the former), yet it may be satisfactory to deduce this last result by a more purely quaternionic method, which can easily be done as follows. The values (4.) of p' and p" give,
XXI... Vp'p"Snp = pSop" - oSpp" = pSpa + p2σ
= (1 − p')Sor + oSp'T = 7S07 + Vrp'o ||| T, VTp'o,
in which p'o denotes a vector p' (because Sp'σ = 0), and ||| ŋ, p' (because Sno'p'o = 0); this line p'o has therefore the direction of the projection of the line ʼn on a plane perpendicular to p', and we are thus led to draw, through the line oc of intersection of the cyclic planes, a plane COD perpendicular to the normal plane to the conic at P, or to let fall (as in Fig. 80) a perpendicular arc CD on the normal arc PD; after which the normal to the sought osculating plane, or the axis OE of the osculating circle sought, as being | Vp'p", will be contained in the plane through the trace T, or OT, or OB, which is perpendicular to the plane of r and p'o, or to the plane DOB ; and therefore the spherical angle DBE (or DAE) will be a right angle, as before.
(7.) We may also observe that if K be the centre of the osculating circle, considered in its own plane, or the foot of the perpendicular on that plane from o, then by XXI.,