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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 pi = pa, and equation (45) becomes

p = pi = P2; (56)

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 /, m, n; but this is effected if

it = K = L; (57)

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)

Q

P = ,

H

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 1, arc rendered indeterminate, we may proceed as follows:

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U = V = W = 0;

_ 2 2 2

It is evident therefore that H, K, L can never be equal. Also of conditions (61), (62) and (63), the first and last lead to impossible results; and as to (62), let v = 0, therefore y = 0, and

J. if! f!i _ _f!_ *'

AM c* + oM ~ «4ca + c4a2'
as —A2 A2 —ez

~2 _ a2 ,2 _ ,.2 £ £. .

The ellipsoid therefore has four umbilics situated symmetrically in the plane of the greatest and least principal axes; and the tangent planes at these points are parallel to the circular plane sections of the surface. The radius of curvature at the A3

umbilic =

ac

If a = b, the surface becomes an oblate spheroid; x = 0, z = + c, and the umbilics are the points where the axis of revolution meets the surface.

Every point of a sphere is an umbilic; and a sphere is the only surface possessing this property.

Ex. 2. Find the umbilic of the surface xyz = *3.

[table]

yz zx xy

.-. H = — —, K = , L = ,

x y 'z

.-. H = K = L, if x = y = z = k.

and the umbilic is at the point (k, k, k).

If the position of an umbilic is determined by the condition (57), then (24) or (25) is identically satisfied independently of the values of /, m, n; and therefore through such an umbilic the number of lines of curvature may be infinite. But if it is determined by either of the conditions (61), (62), or (63), one only, or at all events only a determinate number of lines of curvature will pass through the umbilical point. Thus through the umbilic on the ellipsoid found in Ex. 1, only one line of curvature passes, and corresponding to it we have

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and therefore the line of curvature is in the plane of the a and c semi-axes.

409 ] To return to the consideration of lines of curvature, and of the locus of the centres of principal curvature which is closely allied to them.

A normal at a point of a curved surface is always intersected by two other adjacent normals, drawn in the two principal normal planes which are perpendicular to each other. Now imagine a point to pass from a given point on a surface to the adjacent point in one of its principal normal sections; and on again in the same direction from this point to the infinitesimally adjacent point in its principal normal section; and so on; it is evident that the point passes along a curve described on the surface; and if the same process is carried on for all the points of the surface, it is evident that the surface will be divided into a series of zones or bands of varying width.

Imagine again the moving point to set out from the first point on the surface along its principal normal section in the direction perpendicular to the former path; and that it passes along this path to a second point; and thence to a third and so on; it is evident that it traverses a path which cuts each of the other curves at right angles; and if the same process is performed in reference to other points on the surface, the surface will be divided into a series of bands of varying width the lines of which are perpendicular to those by which the bands of the former series were formed: and thus the surface will be divided into a series of small curvilinear rectangles, all the angles of which are right, the sides of them being portions of lines of curvature of the surface, and these always intersecting at rightangles.

Let us illustrate these propositions by means of a surface of revolution; say, to fix the ideas, of a paraboloidal form. The lines of intersection of the surface by its meridianal planes evidently form one series of lines of curvature; because all normals along any of these meridianal curves pass through the axis of the figure. And all "parallels" intersect the surface in lines of curvature of the other series, because all the normals drawn at points on the same parallel meet in the axis. The surface-elements into which the whole surface is divided by these lines of curvature are rectangular. If these lines of curvature are projected on a plane perpendicular to the axis of revolution of the surface, the parallels give concentric circles, and the meridians give right lines radiating from the common centre, and therefore cut all the concentric circles at rightangles.

Let us for an instant consider the lines of curvature of the ellipsoid, whose equation is

4. *1 4. f! - 1 as + b2 + c* ~

From (7), Art. 398, we have

{b'l-c2) i + (c2-fl2) Ty + {a'-^ fz = °- (64)

and this is the differential equation to the lines of curvature; from which however wc have at present no means of deducing the integral equation: and therefore we must defer the discussion to the succeeding volume*.

The preceding properties of the lines of curvature, and of the developable surfaces formed by the consecutive normal planes, are of great importance in architecture. If an area is to be covered with a vaulted dome of stone or similar material, the plane joints of the voussoirs ought to be at right angles to the exposed surface of the stone; and therefore the lines of division of the stones in the vault ought to be lines of curvature of the vaulted dome; and then these side-surfaces of the voussoirs will be perpendicular to the exposed surfaces, and the pressures will be at right-angles to the surfaces which bear them. To the observations made by Monge as to the lines of joints of the voussoirs in a vault, I may add, that not only in vaulted domes, but in all kinds of curved masonry and woodwork the lines of joint ought to be the lines of curvature, and the surface joints of the voussoirs ought to be the developable surfaces which correspond to these lines. It is important therefore that we should be able to construct these lines of curvature; aud with this object M. Ch. Dupin discovered the following general theorem:

* For a graphic description of the lines of curvature on an ellipsoid I must refer the reader to " Application de l'Analyse a G&me'trie" of Monge. Price, Vol. 1. 4 r

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