The analytical process for finding the equation of this surfacelocus of two sheets is obvious enough. The equations to the surface, to its lines of curvature through a given point, and to the points of intersection of consecutive normals, or the coordinates to the centre of curvature given by equations (11), are sufficient for eliminating xyz, and for giving an equation in terms of £ ?; and f. As at an umbilic the equations (8) or (24) for determining the directions of the lines of curvature give indeterminate results, we must evaluate them by differentiation according to the method of Art. 139. On examining which it will be seen that each differentiation increases by unity the power of dx, dy, dz; and therefore as we begin with a quadratic, after one differentiation we shall have a cubic; after two differentiations a biquadratic; and so on. Suppose then that the directions are determinate after one differentiation, if all the three roots are real there will be three lines'of curvature passing through the point; if two roots are impossible, there will be but one line of curvature; and such is the case at the umbilics on the ellipsoid. Similarly may there be four or more lines of curvature at an umbilic; nay, an infinite number, as is the case at the pole of a surface of revolution. The lines of curvature are generally non-plane curves, and have a contact of only the first order with the principal normal section, which they touch at the point of contact: this is manifest from the fact that the equation of the lines of curvature involves differentials of the first order only. 413.] If the equation to the surface is given in the explicit f°rm *=/<»,»), (85) we must replace as follows; and as the results assume forms from which most of the properties of the curvature of surfaces have been derived in former text books, we give them in order to exhibit the identity of the conclusions: F (x, y, z) = f(x, y) - z = 0. (86) Let (|) = 9' \dx*> ~ r' \dx~d^i~s' Vd&i ~'' •'• v = p, v = q, w = u = r, v = t, w = 0, I (87) u = 0, v — 0, w = s; J also since dz = pdx + qdy, (88) therefore equation (8) of the lines of curvature becomes dx2 {s (1 4 P2) -pqr} + dydx {t (1 + f) - r (1 + q1)} -dy*{8(l + q*)-pqt} = 0. (89) This is a quadratic equation in terms of dy : dx, and is the differential equation to the projections of the lines of curvature on the plane of xy. Suppose the coordinate planes to be so chosen that the tangent plane at the point under consideration is parallel to the plane of xy, then p = 0, q = 0, and equation (89) becomes , . , , and as the product of the two roots — — 1, it follows that the lines of curvature are perpendicular to each other. Hence it follows, that there is only one line of curvature through an umbilic of the ellipsoid, and that it is in the section of the greatest and least axes. The equation (15) of the principal radii of curvature becomes _ q+^+g')* . (91) the conditions (49), (50), (54) become rt-s2 >, <, = 0. (92) Hence the condition of a developable surface is, see Art. 360, rt-s2 = 0. (93) And the condition (55), that the two principal radii of curvature should be equal and affected with opposite signs, is (1 + q2) r - 2pqs + (1 + jo2) / = 0. (94) Also the general conditions (57) for an umbilic become r a t 1+7 ~ pq ~ i+i5'' (' and of the three sets of special conditions, (61) and (62) become p = 0, * = 0, r(l + 92) = t, (96) q = 0, s = 0, t(l+Pi)=r; (97) (63) cannot be satisfied, since w cannot vanish. Hence also we have a geometrical interpretation of Lagrange's condition that a function of two variables, say z — f{x,y), should admit of a maximum or of a minimum value. The condition is that, see Art. 158 and 159, id2z\ /d2z\ I d2z or rt — a2, should be positive; whence it follows, that the principal radii of curvature must be measured in the same direction. Now at a point where = = 0, the tangent plane is parallel to that of xy; if then all the radii of curvature of the normal sections at that point are measured in the same direction, z is a maximum or a minimum: but if some are turned in one direction and others in the opposite direction, that is, if rt — s2 is negative, there will be a partial maximum and a partial minimum, but no total maximum or minimum; and if rt — a2 = 0, then the surface is developable, and the generating line will give a series of partial maxima or minima. 414.] Meunier's Theorem on the curvature of oblique sections of a surface. The two cases of principal and of ordinary normal sections having thus been investigated, it remains for us to consider the third case of Art. 397; that in which a curve is described on a surface, but the osculating plane to which at the given point is not normal to the surface. Let p be the radius of absolute curvature of such a curve, and let A, p, v be the direction-angles of its direction; then, taking s to be equicrescent, by (26), Art. 378, U V w multiplying these severally by -, -, —, and adding, we have , u d2x + v d2y + w dH v cos k + v cos n + w cos v r QOS2 Q but the right-hand member of the equation is the cosine of the angle between the radius of curvature of the oblique section and the normal to the surface, = cos \jr, say; and by equation (12), if p is the radius of curvature of the normal section at PRICE, Vol. i. 4 G the point, and which has the same tangent, the latter factor of the left-hand member = P .-. p = pcosfl. (100) Hence the radius of curvature of an oblique section is equal to the projection on the osculating plane at the point, of the radius of curvature of the normal section of the surface which has the same tangent line with the oblique section. Hence if a sphere is described having for centre and radius the centre and radius of curvature of any normal section, all the oblique sections which touch the normal section at the point on the surface have, for osculating circles at the common point, the small circles of the sphere made by their respective planes. 415.] As whatever tends to elucidate the difficulties of an obscure subject deserves attention, I do not hesitate to introduce the following process, although it proves theorems which have been discussed in the previous Articles; and it exhibits the relations existing between the curvatures of normal sections in a remarkable light, and hereby indicates the nature of a point of a surface at which the partial derived-functions are not indeterminate. Let the point of the surface under consideration be taken as the origin, and let the tangent plane be that of xy; and therefore the normal is the axis of z. Let the equation to the surface be *=/(•,*). (101) At an infinitesimal distance dz from the origin let a plane be drawn parallel to that of xy, and cutting the surface; the curve of section we will, after M. Ch. Dupin,* call the indicatrix, as the form of it indicates the nature of the surface at the origin; let dx, dy, dz be the coordinates to a point on this curve; and through that point and the axis of z let a normal section be drawn, making an angle a with the plane of xz, so that da tan a = ~; and let ds be the arc of the normal section of the dw surface between the origin and the point (dx. dy, dz); then ds2 = da? + dy2 + dz*; (102) and if p is the radius of curvature at the origin ol this normal * Developpements de Ge'ome'trie; par Ch. Dupin, Paris 1813, page 48. section, and which lies along the axis of z, from the geometry of the circle, we have , 2 "= db; <103> that is, the radius of curvature of a normal section varies as the square of the distance between the point and the intersection of the normal plane with the indicatrix. Using the notation of Art. 413, and expanding according to Art. 140, we have z + dz = z+pdx + qdy + Y-^{rdx2 + 2sdxdy + tdy2} + (104) and neglecting higher powers of the infinitesimals dx and dy, and observing that p = q = 0, because the normal at the origin is perpendicular to the axes of x and y, we have dz = g {rdx2 + 2sdxdy + tdy2}; (105) and therefore 9 = rdxt + lsdxdy + tdy*' (106) which equation is equivalent to (15), and gives the value of the radius of curvature of the normal section. As p is generally finite, it appears from equation (103) that dz is an infinitesimal of the same order as ds2; therefore in equation (102) dz% must be neglected, and we have ds2 = da* + dy2. (107) HeDCe P = rdx*+tdx%+tdy2> <108> — r (cos a)2 + 2s sin a cos a +1 (sin a)2' ^9) which result is the same as equation (91), and from which therefore the properties of maxima and minima radii of curvature might be deduced. Suppose the coordinate axes of x and y to be turned about the axis of z through an angle 6, such that tan20=-^7; (110) T — I then (105) becomes 2 dz = r dx2 + tdy2; (111) which equation, if we consider dx, dy, dz to be the Coordinates to a point on the surface near to the origin, is that of a para |