the osculating plane, and is called the absolute (or principal) normal; and the one which is perpendicular to that plane, and which it has been lately proposed to name the binormal.* It is easy to assign expressions, by quaternions, for these two normals, as follows. (1.) The absolute normal, as being perpendicular to p', but complanar with p’ and p”, has a direction expressed by any one of the following formula (comp. 203, 334): I... Vp"p'.p'-1; or II... dUp'; or III... dUdp. (2.) There is an extensive classt of cases, for which the following equations hold good: IV... Tp' const. ; V... p'2 = const.; VI... Sp'p"=0; and in all such cases, the expression I. reduces itself to p", which is therefore then a representative of the absolute normal. (3.) For example, in the case of the helix, with the equation several times before employed, the conditions (2.) are satisfied; and accordingly the absolute normal to that curve coincides with the normal p" to the cylinder, on which it is traced; the locus of the absolute normal being here that screw surface or Helicoid, which has been already partially considered (comp. 314, (11.), and 372, (4.)). (4.) And as regards the binormal, it may be sufficient here to remark, that because it is perpendicular to the osculating plane, it has the direction expressed by one or other of the two symbols (comp. 377, V.), VII... Vo'p", or VII... Vdpd2p. (5.) There exists, of course, a system of three rectangular planes, the osculating plane being one, which are connected with the system of the three rectangular lines, the tangent, the absolute normal, and the binormal, and of which any one who has studied the Quaternions so far can easily form the expressions. (6.) And a construction‡ for the absolute normal may be assigned, analogous to and including that lately given (378) for the osculating plane, as an interpretation of the expression II. or III., or of the symbol dUp' or dUdp. From any origin o conceive a system of unit lines (Up' or Udp) to be drawn, in the directions of the successive tangents to the given curve of double curvature; these lines will terminate *By M. de Saint-Venant, as being perpendicular at once to two consecutive elements of the curve, in the infinitesimal treatment of this subject. See page 261 of the very valuable Treatise on Analytic Geometry of Three Dimensions (Hodges and Smith, Dublin), by the Rev. George Salmon, D. D., which has been published in the present year (1862), but not till after the printing of these Elements of Quaternions (begun in 1860) had been too far advanced, to allow the writer of them to profit by the study of it, so much as he would otherwise have sought to do. Namely, those in which the arc of the curve, or that arc multiplied by a scalar constant, is taken as the independent variable. This construction also has not been met with by the writer in print, so far as he remembers; but it may easily have escaped his notice, even in the books which he has seen. CHAP. III.] ABSOLUTE NORMALS, GEODETIC LINES. 515 on a certain spherical curve; and the tangent, say ss”, to this new curve, at the point B which corresponds to the point P of the old one, will have the direction of the absolute normal at that old point. (7.) At the same time, the plane oss' of the great circle, which touches the new curve upon the unit sphere, being the tangent plane to the cone of parallels (378), has the direction of the osculating plane to the old curve; and the radius drawn to its pole is parallel to the binormal. (8.) As an example of the auxiliary (or spherical) curve, constructed as in (6.), we may take again the helix (369, XIII., &c.) as the given curve of double curvature, and observe that the expression 369, XIV., namely, whence Tp' is constant (as in IV.), and we have the equation (comp. 369, XV. XIX.), :) = cos a const., a being again the inclination of the helix to the axis of its cylinder; which shows that the new curve is in this case a plane one, namely a certain small circle of the unit sphere. (9.) In general, if the given curve be conceived to be an orbit described by a point, which moves with a constant velocity taken for unity, the auxiliary or spherical curve becomes what we have proposed (100, (5.)) to call the hodograph of that motion. (10.) And if the given curve be supposed to be described with a variable velocity, the hodograph is still some curve upon the cone of parallels to tangents. SECTION 5.-On Geodetic Lines, and Families of Surfaces. 380. Adopting as the definition of a geodetic line, on any proposed curved surface, the property that it is one of which the osculating plane is always a normal plane to that surface, or that the absolute normal to the curve is also the normal to the surface, we have two principal modes of expressing by quaternions this general and characteristic property. For we may either write, to express that the normal v to the surface (comp. 373) is perpendicular to the binormal Vp'p" or Vdpd2p to the curve (comp. 379, VII. VII'.); or else, at pleasure, to express that the same normal has the direction of the absolute normal (Up') or dUdp (comp. 379, II. III.), to the same geodetic line. And thus it becomes easy to deduce the known relations of such lines (or curves) to some important families of surfaces, on which they can be traced. Accordingly, after beginning for simplicity with the sphere, we shall proceed in the following sub-articles to de-. termine the geodetic lines on cylindrical and conical surfaces, with arbitrary bases; intending afterwards to show how the corresponding lines can be investigated, upon developable surfaces, and surfaces of revolution. (1.) On a sphere, with centre at the origin, we have v || p, and the differential equation IV. admits of an immediate integration ;* for it here becomes, V... 0=VpdUdp = dVpUdp, whence VI... VpUdp=w, and VII. . . Swp = 0, being some constant vector; the curve is therefore in this case a great circle, as being wholly contained in one diametral plane. (2.) Or we may observe that the equation, VIII... Spp'p" =0, or IX... Spdpd2p = 0, obtained by changing v to p in I. or II., has generally for a first integral (comp. 335, (1.)), whether To be constant or variable, it expresses therefore that p is the vector of some curve (or line) in a plane through the origin; which curve must consequently be here a great circle, as before. (3.) Accordingly, as a verification of X., if we write XI... p = ax +ẞy, x and y being scalar functions of t, where t is still some independent scalar variable, and a, ẞ are two vector constants, we shall have the derivatives, XII. . . p' = ax' +ẞy', p" = ax" + ẞy" ||| p, p' ; so that the equation VIII. is satisfied. (4.) For an arbitrary cylinder, with generating lines parallel to a fixed line a, we may write, XIII... Sav = 0, XIV... SadUdp = 0, XV... SaUdo = const. ; a geodetic on a cylinder crosses therefore the generating lines at a constant angle, and consequently becomes a right line when the cylinder is unfolded into a plane: both which known properties are accordingly verified (comp. 369, (5.), and 376, (2.)) for the case of a cylinder of revolution, in which case the geodetic is a helix. (5.) For an arbitrary cone, with vertex at the origin, we have the equations, XVI... Svp = 0, XVII... SpdUdp = 0, XVIII. . . dSpUdp = $(dp.Udp) = — Tdp ; multiplying the last of which equations by 2SpUdp, and observing that - 2Spdp =-d. p2, we obtain the transformations, * We here assume as evident, that the differential of a variable cannot be constantly zero (comp. 335, (7.)); and we employ the principle (comp. 338, (5.)), that V.dp Udp=- VTdp = 0. CHAP. III.] GEODETICS ON SPHERres, cones and CYLINDERS. 517 XIX...0=d{(SpUdp)2 + p2} = d. (VpUdp)*, XX... TVpUdp = const. ; the perpendicular from the vertex, on a tangent to any one geodetic upon a cone, has therefore a constant length; and all such tangents touch also a concentric sphere,* or one which has its centre at the vertex of the cone. (6.) Conceive then that at each point P or P' of the geodetic a tangent PT or P'T' is drawn, and that the angles oTP, OT'P' are right; we shall have, by what has just been shown, XXI... OTOT = const. = radius of concentric sphere; P and if the cone be developed (or unfolded) into a plane, this constant or common length, of the perpendiculars from o on the tangents, will remain unchanged, because the length OP and the angle OPT are unaltered by such development; the geodetic becomes therefore some plane line, with the same property as before; and although this property would belong, not only to a right line, but also to a circle with o for centre (compare the second part of the annexed Figure 78), yet we have in this result O merely an effect of the foreign factor SpUdp, which was introduced in (5.), in order to facilitate the integration of the differential equation XVIII., and which (by that very equation) cannot be constantly equal to zero. We are therefore to exclude the curves in which the cone is cut by spheres concentric with it: and there remain, as the sought geodetic lines, only those of which the developments are rectilinear, as in (4). P. T T=T' 'T Fig. 78. (7.) Another mode of interpreting, and at the same time of integrating, the equation XVIII., is connected with the interpretation of the symbol Tdp; which can be proved, on the principles of the present Calculus, to represent rigorously the differential ds of the arc (s) of that curve, whatever it may be, of which p is the variable vector; so that we have the general and rigorous equation, XXII. . . Tdp = ds, if s thus denote the arc : whether that are itself, or some other scalar, t, be taken as the independent variable; and whether its differential ds be small or large, provided that it be positive. (8.) In fact if we suppose, for the sake of greater generality, that the vector p and the scalars are thus both functions, pt and st, of some one independent and scalar variable, t, our principles direct us first to take, or to conceive as taken, a submultiple, n-1dt, of the finite differential dt, considered as an assumed and arbitrary increment of that independent variable, t; to determine next the vector Pt ̄1dt, and the scalar st+n-1dt, which correspond to the point Pt+n ̄1at of the curve on which p; terminates in Pt, and of which s is the arc, PPt, measured to P from some fixed point Po on the same curve; to take the differences, When the cone is of the second order, this becomes a case of a known theorem respecting geodetic lines on a surface of the same second order, the tangents to any one of which curves touch also a confocal surface. which represent respectively the directed chord, and the length, of the arc PPL, which arc will generally be small, if the number ʼn be large, and will indefinitely diminish when that number tends to infinity; to multiply these two decreasing differences, of pt and st, by n; and finally to seek the limits to which the products tend, when n thus tends to oo: such limits being, by our definitions, the values of the two sought and simultaneous differentials, dp and ds, which answer to the assumed values of t and dt. And because the small arc, As, and the length, TAP, of its small chord, in the foregoing construction, tend indefinitely to a ratio of equality, such must be the rigorous ratio of ds and Tdp, which are (comp. 320) the limits of their equimultiples. (9.) Admitting then the exact equality XXII. of Tdp and ds, at least when the latter like the former is taken positively, we have only to substitute - ds for – Tdp in the equation XVIII., which then becomes immediately integrable, and gives, XXIII. . . s + SpUdp = s- where S (p: Udp) denotes the projection TP, of the vector p or OP, on the tangent to the geodetic at P, considered as a positive scalar when p makes an acute angle with do, that is, when the distance Tp or OP from the vertex is increasing; while s denotes, as above, the length of the arc PoP of the same curve, measured from some fixed point Po thereon, and considered as a scalar which changes sign, when the variable point P passes through the position Po (10.) But the length of TP does not change (comp. (6.)), when the cone is developed, as before; we have therefore the equations (comp. again Fig. 78), FOR-T'F', XXIV... POP-TP= const. = POP - T'P', XXV... PP' = T'P' — TP, which must hold good both before and after the supposed development of the conical surface; and it is easy to see that this can only be, by the geodetic on the cone becoming a right line, as before. In fact, if or' in the plane be supposed to intersect the tangent TP in a point r', and if p' be conceived to approach to P, the second member of XXV. bears a limiting ratio of equality to the first member, increased or diminished by TT.; which latter line, and therefore also the angle TOT' between the perpendiculars on the two near tangents, or the angle between those tangents themselves, if existing, must bear an indefinitely decreasing ratio to the arc PP'; so that the radius of curvature of the supposed curve is infinite, or T' coincides with T, and the development is rectilinear as before. (11.) The important and general equation, Tdp = ds (XXII.), other consequences, and may be put under several other forms. may write generally, also XXVI... TDsp = 1, XXVII... (Dsp)2 + 1 = 0; conducts to many For example, we XXVIII. . . (Dep)2 +(Dis)2=0, or XXIX... p22 + s′2 = 0, if p' and s' be the first derivatives of p and s, taken with respect to any independent scalar variable, such as t; whence, by continued derivation, XXX... Sp'p" + s's" =0, XXXI... Sp'p" + p"2+s's"" + s′′2= 0, &c. (12.) And if the arc s be itself taken as the independent variable, then (comp. 379, (2.)) the equations XXIX., &c., become, XXXII. . . p2+1=0, Sp'p" =0, Sp'p" + p12 = 0, &c. |