desic line is drawn from one curve to another curve on the sur the geodesic line, and dæ, dy, dz are proportional to the directioncosines of the tangent to the limiting curve at the limit, it appears that the geodesic line cuts both the limiting curves at rightangles: this is also manifest by general reasoning. It will be observed however that although an infinitesimal arc of a geodesic is of a minimum length between its extremities, yet if the distance between the extreme points is finite, a geodesic passing through them may be either a maximum or a minimum, or indeed only one of such critical lines, the number of which may be infinite. Of this theorem we shall hereafter have some instances. dy dz d. d. are proportional to the directionds ds cosines of the principal normal, or of the direction of the radius of absolute curvature of a curve in space, and since U, V, w are proportional to the direction-cosines of the normal to the surface at the point (x, y, z), (66) shew that the radius of absolute curvature of a geodesic line drawn on a surface is coincident in direction with the normal to the surface; or, in other and equivalent words, that the osculating plane of a geodesic line is a normal plane to the surface. Hence it appears that if a cylinder or other developable surface touches a given surface along a geodesic, the line of contact becomes straight when the developable surface is unwrapped into a plane. 337.] The equations to a geodesic line on a surface may be put under the following form: Since the osculating plane of the geodesic line contains the normal to the surface, we have x (dy d2z — dz d3y) +v (dz d2x − dx d2z) + w (dx d2y — dy d2x) or (v d2z-wd2y) dx + (w d2x — v d2 z) dy + (v d3y — v d2x) dz = 0, = 0; = = dx2 + dy2 + dz2 Q2 (dx d2x+dy d2y+dz d2z) dvdx+dv dy+dw dz (67) Q2 (dv d2x + dv d2y+dw d2z — (v du+vdv+wdw)(u d2x+vd2y+wd2z)' and since U dx+v dy+wdz = 0, vd2x+vd2y + w d2z = (dvdx + dv dy + dw dz); so that from (67) we have dvd2x+dv d2y+dwd2z v dv+vdv+wdw dx d1x + dy d2y+dzd2z dvdx+dv dy+dwdz or + Q2 =0, dv d2x+dv d2y+dw d2z dq + The element of the integral in the first member of this equation is manifestly a differential of the second order; and the integral will consequently involve two arbitrary constants one of which will depend on, say, the initial direction, and the other on the initial point of the geodesic. The complete integral of (68) has not yet been found. In the case of surfaces of the second degree the first integral can be found, as we shall presently see. 338.] Let p be the radius of absolute curvature of a geodesic line; then by (23), Art. 377, Vol. I, by means of either of which equations the length of the radius of absolute curvature at any point may be determined. Also let p' be the radius of curvature of the normal section of the surface which contains ds; then, from (69), that is, the radius of absolute curvature of a geodesic line is equal to the radius of curvature of the normal section of the surface, which at their common point touches the line. This result may also be inferred from the property stated in Art. 336; viz., the osculating plane of a geodesic line is a normal plane to the surface. Consequently two consecutive elements of the geodesic are in the normal plane, and are coincident with two consecutive elements of the normal curve of section; thus the radius of absolute curvature of both curves is the same. 339.] Also to determine the torsion of a geodesic, let us take the value of the radius of torsion which is given in (41), Art. 382, Vol. I: whereby we have P Qds+ {(dz d3y — dy d3z) v + (dx d3z— dz d3x)v + (dy d3x — dx d3y)w} And since xd2x+yd2y+zd2z = 0, by (69) we have UX+VY+WZ = 0. -(Udx+vdy+wdz) = xdu+Ydv+vdw. (73) (74) (75) 1 {(wdy —vdz) (vdx —vdy) Q2ds2 { (w dy —v dz) du+ (u dz—w dx) dv + (v dx — v dy)dw} (vdw-wdv) dx+(wdu-u dw) dy + (v dv - v dv) dz which gives the radius of torsion at any point of a geodesic. (77) If the numerator of (77) = 0, dx, dy, dz refer to a line of curvature at the point (x, y, z), by reason of (7), Art. 398, Vol. I. Hence it follows that if a geodesic touches a line of curvature, its torsion is suspended at the point of contact. And consequently if a line of curvature is a geodesic, it is a plane curve, because every point on it is a point of suspended torsion. Also since there are two lines of curvature passing through every point and at right angles to each other, there are also at every point two geodesics whose torsion is suspended at that point. 340.] Let us further consider geodesics at a point in reference to the lines of curvature which pass through that point. Let the numerator of (77) be replaced by its equivalent which is given in (24), Art. 399, Vol. I; then U (K—L) dy dz +V (L—H) dz dx + w (H − K) dx dy Q2 ds2 --- (78) Let (1, m, n), (1, m1, n1), (l2, m2, ng), be sets of direction-cosines referring to the geodesic and to the two lines of curvature at the point (x, y, z); let be the angle at which the geodesic in its first element is inclined to the first line of curvature; and let P1 and p2 be the principal radii of curvature at the point; then Ρι P2 U (K — L) m ̧n ̧ +V (L—н) n12+W (н—K)↳ ̧m, = 0, Į (80) and by (41), Art. 403, Vol. I, 1 = ↳1 cos 0 + 11⁄2 sin 0, m = m1cos0+m, sin 0, n = n1 cos 0+n2 sin 0; (81) therefore substituting these in (79), and omitting terms which +V (L−H) (n2 l1⁄2 + l1n2) + W (H − K) (l1M2+M1l1⁄2)} ; + (L—H) (n ̧2 7,2 — l ̧2 n‚2) + (H — K) (l2 m2 — m‚2 1,2)} sin cos 0 {H (l,2 — 1,2)+K(m ̧2 — m2) + L(n ̧2—n‚2)} and this assigns the radius of torsion in terms of the principal radii of curvature, and of the angle between the first element of the geodesic and a line of curvature. ing theorems. Hence we have the follow The torsion is suspended when 0 = 0, and when ✪ when the geodesic touches a line of curvature. 0 The torsion is a maximum when = 45°, and 0 = 135°; that is, when the geodesics bisect the angles between the lines of curvature. Thus if lines of maximum geodesic torsion are traced on a given surface, two such will pass through every point on the surface, and they will bisect the angles between the lines of curvature at that point. The torsions of two geodesics which intersect at right-angles are equal at the point of intersection. If P1 = P2, that is, if the point (x, y, z) is an umbilic, the torsion at that point of every geodesic passing through it is suspended. And conversely, if at a given point the torsion of every geodesic passing through it is suspended, that point is an umbilic. The equation (82) may also be put into the following form. Let ρ and p' be the radii of curvature of the normal sections coincident with and perpendicular to the geodesic; then by Euler's Theorem, (45), Art. 403, Vol. I, 341.] The preceding theory of geodesics leads to some theorems concerning lines drawn on a surface which deserve attention. Let PQR... be a curved line drawn on a surface, of which |