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,

or rt — s2, should be positive; whence it follows, that the principal radii of curvature must be measured in the same direction.

dz
dx

dz

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 — s2 = 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 λ, μ, be the direction-angles of its direction; then, taking s to be equicrescent, by (26), Art. 378,

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 y, 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

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

z = f(x, y).

(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 dy tan a = ; and let ds be the arc of the normal section of the dx

surface between the origin and the point (da, dy, dz); then

ds2 = dx2 + dy2 + dz2 ;

(102)

and if p is the radius of curvature at the origin of this normal

* Développements de Géométrie; par Ch. Dupin, Paris 1813, page 48.

section, and which lies along the axis of z, from the geometry of the circle, we have

ds2

(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

1

z+dz = z+pdx+qdy+ {rdx2+2s dx dy+tdy2} + ..., (104)

1.2

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

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) dz2 must be neglected, and we have

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 0, such that

which equation, if we consider da, dy, dz to be the coordinates to a point on the surface near to the origin, is that of a para

boloid; of which the principal sections are those made by the planes of az and yz; and if pr and p, are the radii of curvature

of these sections respectively, Px =

1

1

› Py =

If the axes of

t

x and y are turned about that of z as above, (109) becomes

which is Euler's Theorem; see Art. 403.

To return to the equation of the indicatrix, viz. (105); dz, being the distance between the parallel planes of ry and of that of the curve, is constant; and do and dy are the rectangular coordinates to the indicatrix, the origin being at the point where the axis of cuts its plane, and ds is the radius vector; hence,

C

replacing dæ and dy by § and ŋ, and dz by 2

r2+2sEn+tn2 = c,

we have

(113) which is an equation of the second degree, referred to its centre as origin, and represents an ellipse or hyperbola according as rt-s2 is positive or negative; and represents a circle if r = t, and s = 0; and two parallel straight lines if rt-s2 = 0. Hence we conclude, that if a surface is cut by a plane parallel and infinitesimally near to a tangent plane, the curve of section is either an ellipse, a hyperbola, or two parallel straight lines; the ellipse of course admitting of the variety of a circle, and the hyperbola in certain cases being rectangular, and in other two intersecting straight lines.

If the indicatrix is an ellipse the surface is wholly concave towards it, such as is the case at all points of an ellipsoid; and, if it is a hyperbola, some part of the surface about the point has its curvature turned in one direction and some part in the opposite; and if the indicatrix is two parallel straight lines, the surface is concave towards them in a direction perpendicular to them, but is in a straight line in a direction parallel to them.

Also since ds is the central radius vector of the indicatrix, and since the radius of curvature of the normal section varies as ds2, the latter quantity partakes of singular values analogous to those which the former admits of.

In the ellipse all the radii vectores are real; therefore if the

indicatrix is an ellipse, that is, if rt-s2 is positive, all the radii of curvature of normal sections are turned in the same direction : the radii vectores of the ellipse have two maxima and two minima values, which are at right angles to each other; therefore the radii of curvature of normal sections have values respectively a maximum and a minimum, which are perpendicular to each other. In the circle all the radii vectores are equal; therefore, if rt and s= 0, all the radii of curvature of normal sections are equal, and there is an umbilic; thus the tangent plane at an umbilic of an ellipsoid is parallel to a plane of circular sections.

In the hyperbola some of the radii vectores are real and some are impossible; therefore if rt-82 is negative, the radii of curvature of normal sections are turned in one direction for all real radii vectores of the hyperbola, and in the opposite direction for the impossible ones; the asymptotes being the lines bounding the parts which have their curvatures turned in opposite directions; and if the hyperbola is rectangular, equal portions of the surface at the point have their curvatures turned in opposite directions. Hence also, as the principal axes of the hyperbola are at right angles, one being real and the other being impossible, so will the sections of greatest and least curvature be at right angles to each other, and the radii will be turned in opposite directions.

=

If rt s2, that is if the indicatrix is two parallel straight lines, the origin being at a middle point between them, the radii vectores which are perpendicular to the lines are the least, and the normal section coincident with them is that of greatest curvature; but as the line, which is parallel to and bisects them, never meets them, the corresponding radius of curvature is infinite, and the curvature of the coincident normal section vanishes. This is manifestly the case with developable surfaces.

416.] Hence also it is plain, that if the condition of oscu lation of two surfaces is made to depend on the second derived functions as well as the first being the same in both, or in other words, on the two surfaces having the same indicatrix, a surface of the second order can always be found to osculate to a given surface at a given point; and that, in the case of an umbilic, the surface may be a sphere, and in a developable surface it becomes a cylinder.