« PreviousContinue »
OSCULATING ELLIPTIC CYLINDER.
same given point P, so that we may call it (comp. 397, (31.)) the Osculating Elliptic (or Hyperbolic) Cylinder, perpendicular to the osculating plane.
(36.) As a verification, if we suppress the second member of either LIX. or LIX'., we obtain, under a new form, the equation of what has been already called the Osculating Parabolic Cylinder (397, LXXXIV.); and as another verification, the coefficient of y in that second member vanishes, as it ought to do, when the given curve is supposed to be a parabola : that plane curve, in fact, satisfying the differential equation of the second order,
if r be still the radius of curvature, considered as a function of the arc, s, while p is here the semiparameter.
(37.) The binormal v is, by the construction, a generating line of the cylinder LIX.; and although this line is not generally a side of the cone LVII., yet we can make it such, by assigning the particular value zero to the arbitrary constant, e, in its equation, or by suppressing the term, ez2. And when this is done, the cone LVII. will intersect the cylinder LIX., not only in this common side v (comp. 397, (33.)), but also in a certain twisted cubic, which will have contact of the fourth order with the given curve at P, as stated at the commencement of (34.).
(38.) But, as was also stated there, indefinitely many such cubics can be described, which shall have contact of the same (fourth) order, with the same curve, at the same point. For we may assume any point E of space, or any vector (comp. LVI.),
LXI... OE = = p + ar + brr' + crv,
in which a, b, c are any three scalar constants; and then the vector equation,
LXII... w = ps + t (ε − p),
in which is a new scalar variable, will represent a cylindric surface, not generally of the second order, but passing through the given curve, and having the line PE for a generatrix. We can then cut (generally) this new cylinder by the osculating plane to the curve at P, and so obtain (generally) a new and oblique projection of the curve upon that plane; the x and y of which new projected curve will depend on the arcs of the original curve by the relations,
LXIII. . . x = x,- ac ̄1z, y=ys-bc-1zs;
with the approximate expressions LVIII. for xy s7 8. And if we then determine two new scular constants, B and C, by the condition that the substitution of these last expressions LXIII. for x and y shall satisfy this new equation,
LXIV... 2ry = x2 + 2Bxy + Cy2,
if only s3 be neglected (comp. (35.)), or by equating the coefficients of s3 and s1, in the result of such substitution, then, on restoring the significations LV. of xyz, and writing for abridgment,
LXV... X=x-ac1z, Y=y-bc 1z,
the equation of the second degree,
LXVI... 2r Y= X2 + 2BXY + CY2,
will represent generally an oblique osculating elliptic (or hyperbolic) cylinder, which has contact of the fourth order with the given curve at P, and contains the assumed line PE. If then we determine finally the constant e in LVII., by the result of the substitution of abc for xyz, or by the condition,
the cone LVII., and the cylinder LXVI., will have that line PE for a common side; and will intersect each other, not only in that line, but also (as before) in a twisted cubic, although now a new one, which will have the required (fourth) order of contact, with the given curve at the given point.
(39.) If, after the substitution (38.) in LXIV., we equate the coefficients of the three powers, s3, 84, 85, and then eliminate B and C, we are conducted to an equation of condition, which is found to be of the form,
LXVIII... ab3 + bb2c + cbc2 + ec3: = ac(bg+ch);
in which the ratios of abc still serve to determine the direction of the generating line PE, while the coefficients a, b, c, e, g, h are assignable functions of r, r, r′, r', r′′, r′′, and r", depending on the vector : and when this condition LXVIII. is satisfied, the cylinder LXVI. has contact of the fifth order with the given curve at P.
(40.) Again, if we improve the approximate expressions LVIII. for the three 85714 scalars x, y, zs, by taking account of s, or by introducing the new term 120 (comp. I.) of p., and if we substitute the expressions so improved, instead of x, y, z, in the equation of the cone LVII. and then equate to zero (comp. (34.)) the coefficient of s in the difference of the two members of that equation, we obtain a definite expression for the constant, e, which had been arbitrary before, but becomes now a given function of rrr'r'r" and r" (not involving r"), namely the following
ri 9 21 r'2 3r" 3r'r' 27r'2 9r"
LXIX. . ..e=
+3 735 4r2r2 r2r
and when the constant e receives this value,* the cone has contact of the fifth order with the curve at the given point.
(41.) Finally, if we multiply the equation LXVII. by bg+ch, we can at once eliminate a by LXVIII., and so obtain a cubic equation in b: C, which has at least one real root, answering to a real system of ratios a, b, c, and therefore to a real direction of the line PE in (38.). It is therefore possible to assign at least one real cylinder of the second order (39.), which shall have contact of the fifth order with the curve at P, and shall at the same time have one side PE common with the cone of the second order (40.), which has contact of the same (fifth) order with the curve (or of the fourth order with the cone of chords): and consequently it is possible in this way to assign, as the intersection of this cylinder with this cone, at least one real
* Compare the first Note to page 588.
OSCULATING TWISTED CUBIC.
twisted cubic, which has contact of the fifth order with the given curve of double curvature, at the given point thereof. And such a cubic curve may be called, by eminence, an Osculating† Twisted Cubic.
(42.) Not intending to return, in these Elements, on the subject of such cubic curves, we may take this occasion to remark, that the very simple vector equation,‡ LXX... Ταρ = ρνβρ,
represents a curve of this kind, if a and ẞ be any two constant and non-parallel vectors. In fact, if we operate on this equation by the symbol S. A, in which λ is an arbitrary but constant vector, the scalar equation so obtained, namely,
LXXI... Sλap = SXpSẞp - p2SẞX,
represents a surface of the second order, on which the curve is wholly contained; making then successively λ= a and λ = ß, we get, in particular, the two equations, LXXII... S(Vap.Vßp)=0, and LXXIII... (Vẞp)2 + Saßp = 0, representing respectively a cone and cylinder of that order, with the vector 3 from the origin as a common side: and the remaining part of the intersection of these two surfaces, is precisely the curve LXX., which therefore is a twisted cubic, in the known sense already referred to.
(43.) Other surfaces of the same order, containing the same curve, would be obtained by assigning other values to A; for example (comp. 397, (31.)), we should get generally an hyperbolic paraboloid from the form LXXI., by taking λ ↓ ß. But it may be more important here to observe, that without supposing any acquaintance with the theory of curved surfaces, the vector equation LXX. can be shown, by
Accordingly, it is known (see page 242 of Dr. Salmon's Treatise, already cited), that a twisted cubic can generally be described through any six given points ; and also (page 248), that three quadric cylinders (or cylinders of the second order or degree) can be described, containing a given cubic curve, their edges being parallel to the three (real or imaginary) asymptotes.
† Compare the first Note to page 563.
This example was given in pages 679, &c., of the Lectures, with some connected transformations, the equation having been found as a certain condition for the inscription of a gauche quadrilateral, or other even-sided polygon, in a given spheric surface (comp. the sub-articles to 296): the 2n successive sides of the figure being obliged to pass through the same even number of given points of space. It was shown that the curve might be said to intersect the unit-sphere (p2 = − 1) in two imaginary points at infinity, and also in two real and two imaginary points, situated on two real right lines, which were reciprocal polars relatively to the sphere, and might be called chords of solution, with respect to the proposed problem of inscription of the polygon; and that analogous results existed for even-sided polygons in ellipsoids, and other surfaces of the second order: whereas the corresponding problem, of the inscription of an odd-sided polygon in such a surface, conducted only to the assignment of a single chord of solution, as happens in the known and analogous theory of polygons in conics, whether the number of sides be (in that theory) even or odd. But we cannot here pursue the subject, which has been treated at some length in the Lectures, and in the Appendices to them.
quaternions, to represent a curve of the third degree, in the sense that it is cut, by an arbitrary plane, in three points (real or imaginary). In fact, we may write the equation as follows,
LXXIV... Vqp=-a, if LXXV. . . q=g + ß,
q being here a quaternion, of which the vector part ß is given, but the scalar part g is arbitrary; and then, by resolving (comp. 347) this linear equation LXXIV., we may still further transform it as follows,
LXXVI... g(ga — ẞ2)p = BSẞa + gVẞa – g2a,
which conducts to a cubic equation in g, when combined with the equation,
of any proposed secant plane.
LXXVII... Sεp = e,
(44.) The vector equation LXX., however, is not sufficiently general, to represent an arbitrary twisted cubic, through an assumed point taken as origin; for which purpose, ten scalar constants ought to be disposable, in order to allow of the curve being made to pass through five* other arbitrary points: whereas the equation referred to involves only five such constants, namely the four included in Ua and Uẞ, and the one quotient of tensors Tẞ: Ta (comp. 358).
(45.) It is easy, however, to accomplish the generalization thus required, with the help of that theory of linear and vector functions (pp) of vectors, which was assigned in the Sixth Section of the preceding Chapter (Arts. 347, &c.). We have only to write, instead of the equation LXX., this other but analogous form which includes it,
LXXVIII... Vap + Vpop = 0, or LXXVIII'... pp + cp = a, and which gives, by principles and methods already explained (comp. 354, (1.)), the transformation,
LXXIX... p= ($ + c) ̄1a =
4a + cxa + c2a
a, a, and xa being here fixed vectors, and m, m', m" being fixed scalars, but e being an arbitrary and variable scalar, which may receive any value, without the expression LXXIX. ceasing to satisfy the equation LXXVIII.
*Compare the first Note to page 591. In general, when a curve in space is supposed to be represented (comp. 371, (5.)) by two scalar equations, each new arbitrary point, through which it is required to pass, introduces a necessity for two new disposable constants, of the scalar kind: and accordingly each new order, say the nth, of contact with such a curve, has been seen to introduce a new vector, D,*p, or (-1), subject to a condition resulting from the general equation TDsp = 1, or 21 (comp. 380, XXVI., and 396, III.), but involving virtually two new scalar constants. Thus, besides the four such constants, which enter through and 'into the determination of the directions of the rectangular system of lines, tangent, normal, and binormal (comp. 379, (5.), or 396, (2.)), and of the length of the radius of (first) curvature, r, the three successive derivatives, r', r", r'", of that radius, and the radius r of second curvature, with its two first derivatives, r' and r", have been seen to enter, through the three other vectors, r", 7", v, into the determination (41.) of the osculating twisted cubic.
CHAP. III. VECTOR EQUATION of a twisted CUBIC.
(46.) The curve LXXVIII. is therefore cut (comp. (43.)) by the plane LXXVII. in three points (real or imaginary), answering to and determined by the three roots of the cubic in c, which is formed by substituting the expression LXXIX. for p in the equation of that secant plane; and consequently it is a curve of the third degree, the three (real or imaginary) asymptotes to which have directions corresponding to the three values of c, obtained by equating to zero the denominator of that expression LXXIX., or by making M=0, in a notation formerly employed: so that they have the directions of the three lines ß, which satisfy this other vector equation (comp. 354, I.),
LXXX... Vẞoß = 0.
(47.) Accordingly, if ẞ be such a line, and if y be any vector in the plane of a and ẞ, the curve LXXVIII. is a part of the intersection of the two surfaces of the second order,
LXXXI... Sappp = 0, and LXXXII... Syap + Sypop = 0, whereof the first is a cone, and which have the line ẞ from the origin for a common side (comp. (42.)): the curve is therefore found anew to be a twisted cubic.
(48.) And as regards the number of the scalar constants, which are to be conceived as entering into its vector equation LXXVIII., when we take for øp the form Vqop+ Vλpμ assigned in 357, I., in which go is an arbitrary but constant quaternion, such as g + y, and λ, μ are constant vectors, the term gp of op disappears under the symbol of operation V.p, and the equation (45.) of the curve becomes,
LXXXIII... Vap + pVyp + VpVλpμ=0;
in which the four versors, Ua, Uy, UX, Uμ, introduce each two scalar constants, while the two tensor quotients, Ty: Ta and TAμ: Ta, count as two others: so that the required number of ten such constants (44.) is exactly made up, the curve being still supposed to pass through an assumed origin, and therefore to have one point given. It is scarcely worth observing, that we can at once remove this last restriction, by merely adding a new constant vector to p, in the last equation, LXXXIII.
(49.) Although, for the determination of the osculating twisted cubic (41.), to a given curve of double curvature, it was necessary (comp. (40.)) to employ the vector or D3p, or to take account of $5 in the vector ps, or in the connected scalars xyz, of (34.), and therefore to improve the expressions LVIII., by carrying in each of them (or at least in the two latter) the approximation one step farther, yet there are many other problems relating to curves in space, besides some that have been already considered, for which those scular expressions LVIII. are sufficiently approximate: or for which the vector expression I. suffices.
(50.) Resuming, for instance, the questions considered in (22.) and (23.), we may throw some additional light on the law of the deviation of a near point P, of the curve, from the osculating sphere at P, as follows. Eliminating n by XXXVI. from XXXV., we find this new expression,
the direction of this deviation from the sphere (R) depends therefore on the sign of the scalar radius r1 (23.) of curvature of the cusp-edge (s) of the polar developable and it is outward or inward (comp. 395, (14.)), according as that cuspedge turns its concavity (comp. XLI.) or its convexity, at the centre s of the oscu