which is Euler's Theorem; and is of importance, as by it the radius of curvature of any normal section is expressed in terms of the principal radii of curvature at the point in question.

404] Hence we have the following proposition as to the radii of curvature of any two normal sections which are perpendicular to each other.

Let p and p' be the radii of curvature of two normal sections perpendicular to each other; then

that is, the sum of the curvatures of any two normal sections perpendicular to each other is constant.

405.] As an example of the preceding formulæ, let us take the ellipsoid whose equation is

if p is the perpendicular from the centre on the tangent plane; see equation (31), Art. 338. Hence equation (27) becomes

the roots of which quadratic equation are the greatest and least principal radii of curvature at any point on the ellipsoid.

And as the last term of the quadratic when written in an

integral form is

a2 62 c2
p1

it follows that the product of the great

est and least radii of curvature is invariable for all points for

and subtracting it from the equation to the ellipsoid, we have

which is the equation to a concentric and confocal surface of

the second order.

Also by means of (26), the directions of the principal normal sections may be determined at any point of the ellipsoid.

406.] We proceed now to consider certain singular values of and P2, and the nature of a point on a surface whereat the singular values exist.

In equations (11) and (12) an ambiguity of sign exists, which is introduced in extracting the root of u2+v2+w2, and therefore p may be affected with a + or a sign; and the same ambiguity of sign continues in (27).

As u, v, w are the same for different normal sections at the same point, and as p is an absolute length of line, it appears from (11) that the change of sign arises from έ—x, n−y, Ŝ−z; and therefore the change of sign implies, that the centres of curvature are for different normal sections situated on different sides of the surface.

P2

With respect to (45) it is to be borne in mind, that P1 and are both taken with the positive sign, and that Q has the same sign, viz. +, in both. From (45) therefore it follows, that if Pi and ρι P2 have the same sign, p has always the same sign as either of them; and that therefore all normal sections have their curvature in the neighbourhood of the point turned in the same direction. The analytical condition derived from equation (27) that this should be the case, is, that

U3KL + V2LH + W2Hк must be positive.

(49)

Also it is manifest that, as p1 and p2 are a maximum and a minimum value of the radii of curvature, p always lies between them.

Again, if the signs of p1 and p2 are different, that is, if

U2KL+V2LH + W2HK is negative,

(50)

the radii of curvature of some normal sections are turned in a direction contrary to that of others, and (45) becomes

then for all values of a, from a to a', and from -a to Ta', the radii of curvature of normal sections are turned in the same direction; and when a = + a', and = π ± a', p = ∞o ; then the normal section becomes a straight line in its consecutive elements which abut at the point, or the curvature is suspended; see Art. 383; and for all values of a outside those limits, the curvature of the normal sections is turned in a contrary direction.

According to our hypothesis pi is the maximum and is the minimum radius of curvature.

In the case in which one of the principal radii of curvature is infinite, say p1 = x,

P =

(sin a)2
P2

the analytical condition of which derived from (27) is

(53)

(54)

and which, when expanded, becomes identical with that determined in Art. 360, equation (62), as the differential equation of developable surfaces. Hence we have the geometrical meaning of this equation. One of the principal normal sections of the surface lies along the straight generating line, and therefore the curvature of this section vanishes.

ρι

P2

Again, suppose the two values of p1 and p2 to be equal and of opposite signs; then the coefficient of the second term of the quadratic (27) must be equal to zero, whereby we have

U2 (K + L) + V2 (L + H) + W2 (H+K) = 0.

(55)

In this case, a' of (52)

=

45°; and therefore of the surface

about the point, the curvature of one-half is turned in one direction, and that of the other half in the opposite direction, and the dividing lines of these districts of opposite curvature are two straight lines passing through the point, and perpendicular to each other.

407.] Lastly, let us consider the case when the principal radii of curvature are equal and have the same sign. Here P1 = p2, and equation (45) becomes

that is, the radii of curvature of all the normal sections at such a point are equal. The point is called an umbilic.

At it (27) when arranged in powers of p is a complete square; whence the condition might be found; but the following process is easier. As all the principal radii of curvature must be equal, the direction-cosines which determine their directions must be indeterminate; and as these are the same as those which determine the lines of curvature, equations (25) or (26) must be satisfied identically and independently of any particular values of l, m, n; but this is effected if

these two equation therefore, together with that to the surface, determine the position of an umbilic. In these however the simultaneous values of xyz must not be such as to make to vanish either u, v or w, for, if so, the process according to which H, K, L were determined in Art. 399 fails. And if the two equations (57) are equivalent to only one, then this, together with the equation to the surface, will determine a line on the surface which is the locus of such umbilical points, and is called the line of spherical curvature.

Also if H = K = L, we have from equation (20)

and all the radii of curvature are equal. And as in this case (49) is satisfied, so all the curvatures of the normal section are in the same direction.

In the case however in which either u, v, or w vanishes, and thereby H, K and L are rendered indeterminate, we may proIceed as follows:

Suppose u to vanish; then returning to equation (8) we have

(v v' — ww') dx2 + ww'dy2 —vv'dz2 + (wv′ — vw') dy dz

+(vw-vu-wu) dz dx + (vu' + wu-we) dx dy = 0. (58)

And since u 0, equation (16) becomes

=

which, for an umbilic, must be satisfied independently of

(61)

Hence to find all the umbilics on a surface, we must first seek the number of points which satisfy the general conditions (57); and also inquire when any and what points satisfy either of the three systems (61), (62), and (63).