« PreviousContinue »
these two projections then, or the vector-tangent KK ́ itself, would suffice to determine r and r', or H and P, and thereby all the affections of the curve which depend on s3, but not on s1.
(39.) We have also the similar triangles (see again Fig. 82),
and the vector equations,
XCV... ▲ KKK ∞ K`KK' ∞ KMS;
XCVI... KK': SM = KK, : SK = KK': KM = KK': PK
=r1 Vector of second curvature (IV.);
whence also result the scalar expressions,
XCVII... tan кSK, = tan KPK' = r1 = Second* Curvature (III.):
this last scalar being positive or negative, according as the rotation KSK, (or KPK') appears to be positive or negative, when seen from that side of the normal plane, towards which the conceived motion (396, (1.)) along the given curve, or the unit tangent +r, is directed.†
(40.) Besides the seven expressions, III., XXVII., L., and XCVII., this important scalar r1 admits of many others, of which the following, numbered for reference as 8, 9, &c., and deduced from formulæ and principles already laid down, are examples: and may serve as exercises in transformation, according to the rules of the present Calculus, while some of them may also be found useful, in future geometrical applications.
(41.) We have then (among others) the transformations:
XCVIII... Second Curvature = r1(seven preceding expressions)
=pr1 cot H= TX cos H = r ̄1r' cot P
= r2Sv'r′ = — Sv'r'-1 = — r2STT'T" = STT'-17"
(8, 9, 10, 11)
(12, 13, 14, 15)
(16, 17, 18, 19)
* In illustration it may be observed, that if ds be treated as infinitely small, and if the line KK' be supposed to represent (not the derivative x', but) the differential vector dx = x'ds, then the projections KK, and KK become dr and rr-ids (comp. XCIII. and XCIV.); while KPK' (in Fig. 82) represents the infinitesimal angle r-'ds, through which the osculating plane (comp. (1.)) revolves, round the tangent to the curve during the change ds of the arc.
+ This direction of + is to be conceived (comp. Fig. 81) to be towards the back of Fig. 82, as drawn, if the scalars and r (and therefore also p) be positive.
CHAP. III.] EXPRESSIONS FOR THE SECOND CURVATURE.
R-1 tan rTr" = R-1 tan L
(40, 41, 42, 43)
(44, 45, 46, 47)
(48, 49, 50)
rd cos 4
PKSL, in the forms 50 and 51, being points of the same gauche quadrilateral as in (18.); and a, in 52 and 53,* denoting any constant vector: while several other varieties of form may be deduced from the foregoing by very simple processes, such as the substitution of Uv for rv, &c., which gives for instance (comp. XI'.), from the form 38, these others,
We may also write, with the significations (10.) of Q1 and Q3, the following expression analogous to L.,
which contains the law of the inflexion of the plane curve, into which the proposed curve of double curvature is projected, on its own rectifying plane: the sign of the scalar, to which this last expression ultimately reduces itself, being determined by the rules of quaternions.
(42.) And besides the various expressions for the positive scalar r2, which are immediately obtained by squaring the foregoing forms, the following are a few others :
XCIX... Square of Second Curvature = r2 = Tr-2
= TX2 — r¬2 = r2ST"T'λ = r2 = r2Tv′2 — r−2p#2
= r2Stv't” — r-£p'2 = p2T7′′2 — p-2 — r-2p12 = R-2 (r1Tr”2—1)
= R-2r4Tv22 = R-2Tk2= R-2 tan2 LTT"
(1, 2, 3)
(4, 5, 6)
(7, 8, 9);
while the important vector 7", besides its two original forms VI., admits of the following among other expressions (comp. XX. XXI.) :
C... "=D,3p (= the two expressions VI.)
= r-2Vλ(o − p) = λr' — r-1r'7' = V'T - P2T
(43.) As regards the general theory (396, (5.), &c.) of emanant lines (n) from curves, it might have been observed that if we write,
This last form 53 corresponds to and contains a theorem of M. Serret, alluded
to in the second Note to page 563.
the equation 396, XXXII. takes the simplified form,
CIII... PH = wo− p = nSn ̈1% = projection of vector Y on emanant ŋ;
for example, when n=v, then 0=r1r, and 2=0, PH=0, or w1 = p, as in (1.); and when ŋ= 7, then 0=v, %= r2, so that the projection PH again vanishes, as in 396, (13.).
(44.) In an extensive class of applications, the emanant lines are perpendicular to the given curve (ŋ7); and since we have, by (43.),
we may write, for this case of normal emanation, the formula,
CV... PH = = projection of vector of curvature (7′) on emanant line (n). square of velocity (TO) of rotation of that emanant
for example, when the emanant (7) coincides with the absolute normal (r′), we have then λ, as in (3.), and the recent formula CV. becomes,
CVI... PH = wo- p = % = 7′TX -2 = r27′ sin2 H = (x − p) sin2 H,
which agrees with the expression XXXVIII.
(45.) And in the corresponding case of tangential emanant planes, by making Srn-0 in the second equation 396, XXXVI., and passing to a second derived equation, we find for the intercept between the point P of the curve, and the point, say R, in which the line of contact of the plane with its own envelope touches the cusp-edge of that developable surface, the expression,
which accordingly vanishes, as it ought to do, when ŋ=v, that is, when the emanant plane Sn (w-p) = 0 coincides with the osculating plane XC.
(46.) Some additional light may be thrown on this whole theory, of the affections of a curve in space depending on the third power of the arc, and even on those affections which depend on higher powers of s, by that conception of an auxiliary spherical curve, which was employed in 379, (6) and (7.), to supply constructions (or geometrical representations) for the directions, not only of the tangent (p') to the given curve, to which indeed the unit-vector (7) of the new curve is parallel, but also of the absolute normal, the binormal, and the osculating plane; while the same auxiliary curve served also, in 389, (2.), to furnish a measure of the curvature of the original curve, which is in fact the velocity* of motion in the new or spherical curve, if that in the old or given one be supposed to be constant, and be taken for unity.
* Accordingly the vector of velocity r', of this conceived motion in the auxiliary curve, is precisely what we have called (389, (4.), comp. 396, VI.) the vector of curvature of the proposed curve in space: and its tensor (Tr') is equal to the reciprocal of the radius (r) of that curvature.
CHAP. III.] CASe of constant RATIO OF CURVATURES.
(47.) We might for instance have observed, that while the normal plane to the curve in space is represented (in direction) by the tangent plane to the sphere, the rectifying plane (as being perpendicular to the absolute normal) is represented similarly by the normal plane to the spherical curve: and it is not difficult to prove that the rectifying line has the direction of that new radius of the sphere, which is drawn to the point (say L) where the normal are to the auxiliary curve touches its own envelope.
(48.) The point L thus determined is the common spherical centre (comp. 394, (5.)) of curvature, of the auxiliary curve itself, and of that reciprocal* curve on the same sphere, of which the radii have the directions (comp. 379, (7.)) of the binormals to the original curve; the trigonometric tangent of the arcual radius of curvature of the auxiliary curve is therefore ultimately equal to a small are of that curve, divided by the corresponding arc of the reciprocal curve (or rather by the latter arc with its direction reversed, if the point L fall between the two curves upon the sphere); and therefore to the first curvature (r ̄1) of the given curve, divided by the second curvature (r-1): and thus we have not only a simple geometrical interpretation of the quaternion equation XI'., but also a geometrical proof (which may be said to require no calculation), of the important but known relation XVII., which connects the ratio (r: r) of the two curvatures, with the angle (H) between the tangent (r) and the rectifying line (X), for any curve in space.
(49.) In whatever manner this known relation (tan H=r: r) has once been established, it is geometrically evident, that if the ratio of the two curvatures be constant, then, because the curve crosses the generating lines of its own rectifying developable (396) under a constant angle (H), that developable surface must be cylindrical or in other words, the proposed curve of double curvature must, in the case supposed, be a geodetic↑ on a cylinder (comp. 380, (4.)). Accordingly the point L, in the two last sub-articles, becomes then a fixed point upon the sphere, and is the common pole of two complementary small circles, to which the auxiliary spherical curve (46.), and the reciprocal curve (48.), in the case here considered, reduce themselves; so that the tangent and the binormal to the curve in space make (in the
The reciprocity here spoken of, between these two spherical curves, is of that known kind, in which each point of one is a pole of the great-circle tangent, at the corresponding point of the other: and accordingly, with our recent symbols, we have not only v=Vrr', but also, Vvv′ = r2V v'v ̄1 = r-2p-17 || 7.
The writer has not happened to meet with the geometrical proof of this known theorem, which is attributed to M. Bertrand by M. Liouville, in page 558 of the already cited Additions to Monge; but the deduction of it as above, from the fundamental property (396) of the rectifying line, is sufficiently obvious, and appears to have suggested the method employ ed by M. de Saint-Venant, in the part (p. 26) of his Memoir sur les lignes courbes non planes, &c., before referred to, in which the result is enunciated. Another, and perhaps even a simpler method, suggested by quaternions, of geometrically establishing the same theorem, will be sketched in the present subarticle (49.); and in the following sub-article (50.), a proof by the quaternion analysis will be given, which seems to leave nothing to be desired on the side of simplicity of calculation.
same case) constant angles, with the fixed radius drawn to that point: and the curve itself is therefore (as before) a geodetic line, on some cylindrical surface.
(50.) By quaternions, when the two curvatures have thus a constant ratio, the equations XI'. and XVI. give,
CVIII... (rλ)' = (Uv + rr-17)' = (rr ̄1)'r = 0,
CIX... rλ = a constant vector;
the tangent (7) makes therefore, in this case, a constant angle (H) with a constant line (rλ): and the curve is thus seen again, by this very simple analysis, to be a geodetic on a cylinder. And because it is easy to prove (comp. XXXI.), that we have in the same case the expression,
CX... r sin2 H= radius of curvature of base,
or of the section of the cylinder made by a plane perpendicular to the generating lines, this other known theorem results, with which we shall conclude the present series of sub-articles: When both the curvatures are constant, the curve is a geodetic on a right circular cylinder (or cylinder of revolution); or it is what has been called above, for simplicity and by eminence, a helix.*
398. When the fourth power (s1) of the arc is taken into account, the expansion of the vector p, involves another term, and takes the form (comp. 397, 1.),
so that the new affections of the curve, thus introduced, depend only on two new scalars, such as r' and ', or r' and R', or H' and P', &c. We must be content to offer here a very few remarks on the theory of such affections, and on the manner in which it may be extended by the introduction of derivatives of higher orders.
* In general, the expression XLIV. for the vector w, of the osculating helix, in which rλ-1 = 7 — X-1T', ρ woλ-27', gives Tw's = 1; so that the deriation (8.) may be considered (comp. (13.)) to be measured from the extremity of an arc of the helix, which is equal in length to the arc s of the curve, and is set off from the same initial point P, with the same initial direction: while wo does not here denote the value of w, answering to s = 0, but has a special signification assigned by the formula XXXVIII. It may also be noted that the conception, referred to in (46.), of an auxiliary spherical curve, corresponds to the ideal substitution of the motion of a point with a varying velocity upon a sphere, for a motion with an uaiform velocity in space, in the investigation of the general properties of curves of double curvature: and that thus it is intimately connected (comp. 379, (9)) with the general theory of hodographs.