directly as the perpendicular from the centre on the tangent plane, and inversely as the square of the semidiameter parallel to the tangent line, a well-known theorem. 329.] By the help of the known properties of the central section parallel to the tangent plane, this theorem gives us all the ordinary properties of the directions of maximum and minimum curvature, their being at right angles to each other, the curvature in any normal section in terms of the chief curvatures and the inclination to their planes, &c., &c., without farther analysis. And when, in a future section, we shew how to find an osculating surface of the second order at any point of a given surface, the same properties will be at once established for surfaces in general. Meanwhile we may prove another curious property of the surfaces of the second order, which similar reasoning extends to all surfaces. The equation of the normal at the point p+ dp in the surface treated in last section is @ = p + dp + xp (p+dp). This intersects the normal at p if (§§ 203, 210) 5.δρφρφδρ = 0, (1) that is, by the result of § 273, if dp be parallel to the maximum or minimum diameter of the central section parallel to the tangent plane. Let σ, and σ be those diameters, then we may write in general 01 where p and q are scalars, infinitely small. If we draw through a point P in the normal at p a line parallel to σ1, we may write its equation @= ρ+αφρ+ σι· The proximate normal (1) passes this line at a distance (see § 203) S.(app-dp)UVσ1¢ (p +dp), or, neglecting terms of the second order, 1 TV σφρ (αρθ.φρσιφσι + aqδ.φρσιφση + 4 δ.σισεφρ). The first term in the bracket vanishes because σ, is a principal vector of the section parallel to the tangent plane, and thus the expression becomes = a Hence, if we take a To, the distance of the normal from the new line is of the second order only. This makes the distance of P from the point of contact Top To, i. e. the principal radius of curvature along the tangent line parallel to σ2. That is, the group of normals drawn near a point of a central surface of the second order pass ultimately through two lines each parallel to the tangent to one principal section, and passing through the centre of curvature of the other. The student may form a notion of the nature of this proposition by considering a small square plate, with normals drawn at every point, to be slightly bent, but by different amounts, in planes perpendicular to its edges. The first bending will make all the normals pass through the axis of the cylinder of which the plate now forms part; the second bending will not sensibly disturb this arrangement, except by lengthening or shortening the line in which the normals meet, but it will make them meet also in the axis of the new cylinder, at right angles to the first. A small pencil of light, with its focal lines, presents this appearance, due to the fact that a series of rays originally normal to a surface remain normals to a surface after any number of reflections and refractions. (See § 315). 330.] To extend these theorems to surfaces in general, it is only necessary, as Hamilton has shewn, to prove that if we write dv = pdp, p is a self-conjugate function; and then the properties of p, as explained in preceding Chapters, are applicable to the question. As the reader will easily see, this is merely another form of the investigation contained in § 317. But it is given here to shew what a number of very simple demonstrations may be given of almost all quaternion theorems. The vector v is defined by an equation of the form dfp = Svdp, where f is a scalar function. Operating on this by another independent symbol of differentiation, d, we have But, as d and are independent, the left-hand members of these equations, as well as the second terms on the right (if these exist at all), are equal, so that we have 331.] If we write the differential of the equation of a surface in the form then it is easy to see that dfp = 2Svdp, f(p+dp)=fp+2Svdp+Sdvdp+ &c., the remaining terms containing as factors the third and higher powers of Tdp. To the second order, then, we may write, except for certain singular points, 0 = 2Svdp+Sdvdp, and, as before, (§ 328), the curvature of the normal section whose tangent line is dp is 1 dv S Tv do 332.] The step taken in last section, although a very simple one, virtually implies that the first three terms of the expansion of fp+dp) are to be formed in accordance with Taylor's Theorem, whose applicability to the expansion of scalar functions of quaternions has not been proved in this work, (see § 135); we therefore give another investigation of the curvature of a normal section, employing for that purpose the formulae of § (282). We have, treating dp as an element of a curve, The curvature is, therefore, since v || p" and Tp' = 1, where is a self-conjugate linear and vector function, whose constants depend only upon the nature of the surface, and the position of the point of contact of the tangent plane; so long as we do not alter these we must consider as possessing the properties explained in Chapter V. Hence, as the expression for Tp" does not involve the tensor of dp, we may put for dp any unit-vector 7, subject of course to the And the curvature of the normal section whose tangent is 7 is (1) If we consider the central section of the surface of the second order we see at once that the curvature of the given surface along the normal section touched by τ is inversely as the square of the parallel radius in the auxiliary surface. This, of course, includes Euler's and other well-known Theorems. 334.] To find the directions of maximum and minimum curvature, we have = max, or min. with the conditions, STOT SVT = 0, TT = 1. By differentiation, as in § 273, we obtain the farther equation T δ.ντφτ = 0. = (1) If be one of the two required directions, 7' TUv is the other, for the last-written equation may be put in the form Hence the sections of greatest and least curvature are perpendicular to one another. We easily obtain, as in § 273, the following equation S.v (+STOT)-1v = 0, whose roots divided by Tv are the required curvatures. 335.] Before leaving this very brief introduction to a subject, an exhaustive treatment of which will be found in Hamilton's Elements, we may make a remark on equation (1) of last section δ.ντφτ = 0, or, as it may be written, by returning to the notation of § 333, This is the general equation of lines of curvature. For, if we define a line of curvature on any surface as a line such that normals drawn at contiguous points in it intersect, then, dp being an element of such a line, the normals must intersect. This gives, by § 203, the condition EXAMPLES TO CHAPTER IX. 1. Find the length of any arc of a curve drawn on a sphere so as to make a constant angle with a fixed diameter. 2. Shew that, if the normal plane of a curve always contains a fixed line, the curve is a circle. 3. Find the radius of spherical curvature of the curve p = pt. Also find the equation of the locus of the centre of spherical curvature. 4. (Hamilton, Bishop Law's Premium Examination, 1854.) (a.) If p be the variable vector of a curve in space, and if the differential de be treated as = 0, then the equation dT (p-k) = 0 obliges κ to be the vector of some point in the normal plane to the curve. (6.) In like manner the system of two equations, where d and d2k are each = dT (p-k) = 0, d2T (p-k) = 0, represents the axis of the element, or the right line drawn through the centre of the osculating circle, perpendicular to the osculating plane. K (c.) The system of the three equations, in which is treated as constant, dT (p-k) = 0, d2T (p−k) = 0, d3T (p-k) = 0, determines the vector κ of the centre of the osculating sphere. (d.) For the three last equations we may substitute the follow ing: S.(p-k) dp = 0, S.(p-k) d2p+dp2 = 0, S.(p-k) d3p+3S.dpd2p = 0. (e.) Hence, generally, whatever the independent and scalar variable may be, on which the variable vector p of the curve depends, the vector κ of the centre of the oscu K lating sphere admits of being thus expressed: K = p + 3V.dpd2pS.dpd2p-dp2 V.dpd3p |