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boloid; of which the principal sections are those made by the planes of xz and yz; and if px and p„ are the radii of curvature

of these sections respectively, px = i, p, = y. If the axes of

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

- r (cos a)1 + t (sin a)2

= (cos^ + (sina)^

Px Py

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 xy and of that of the curve, is constant; and dx and dy are the rectangular coordinates to the indicatrix, the origin being at the point where the axis of z cuts its plane, and ds is the radius vector; hence,


replacing dx and dy by f and rj, and dz by ~ we have


2+2s£T, + tri* = c, (113) which is an equation of the second degree, referred to its centre as origin, and represents an ellipse or hyperbola according as rt—s'1 is positive or negative; and represents a circle if r = r, and s = 0; and two parallel straight lines if rt—s* = 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 f/*z, the latter quantity partakes of singular values analogous to those which the former admits of.

In the ellipse all the radii vectores arc real; therefore if the indicatrix is an ellipse, that is, if rt «2 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 r = / and * = 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—s2 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 osculation 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. •

417.] On the measure of curvature of a surface at a given point.

Gauss in his celebrated memoir "Disquisitiones Generates circa Superficies Curvas," has introduced a definition of curvature of a surface which is derived analogously from the means of measuring the curvature of a plane curve; and from his definition has deduced a mathematical estimate of curvature.

Suppose As to be the finite arc of a plane curve commencing at a point F; aud at the extremities of As let two normals to the curve be drawn. In the same plane take a circle whose radius is unity, and through its centre let two radii be drawn parallel to the two normals at the extremities of As, and let the radii

include an angle or arc A\jr; then the limit towards which


— converges, when the arc of the original curve is infinitesimal, is, according to the definition of Art. 281, the curvature of the curve at the point p.

Imagine now upon a curved surface a finite area enclosed by a contour, within which is a given point p; and also imagine a sphere whose radius is unity; and suppose normals to the surface to be drawn at every point of the enclosing contour, and radii of the sphere to be drawn parallel to these normals; by this process a spherical area will be enclosed on the surface of the sphere.

Let As be the area enclosed by the contour, and As the area of the enclosed figure on the surface of the sphere; then

the limit ^ towards which converges, when the contour

becomes infinitesimal but still encloses the point P, is the curvature of the surface at the point P.

Let the area A 2 on the given surface be a rectangle contained by four lines of curvature; and let A<£1( A fa be the angles subtended at the centres of principal curvature by two adjacent sides of the rectangle;

.-. AS = frptAfaAfa; (114)

and similarly, As = AcpiAfa; (115)

.•. the curvature = -5- = —-—; (116) as px pi

and the curvature of the surface at any point is equal to the product of the curvatures of the principal normal sections at the same point.

The truth of the result is manifestly independent of the form of the small area; for whatever its form be, it can always be divided into a number of infinitesimal rectangles, for every one of which the result of equation (116) will be true; and therefore by simple addition the aggregate of all, which is expressed by equation (116), will be true also. The curvature then will be affected with a positive or a negative sign, according as the radii of the principal normal sections have the same or different signs.

In the case of developable surfaces, one of the principal normal sections has an infinite radius of curvature; it would therefore follow from (116), that the curvature of a developable surface is zero: but such is the case only with a plane. We must therefore retrace our steps and modify the process in the following manner, by operating on a right circular cylinder whose radius is unity instead of on a sphere:

Let the two containing sides of the rectangular area on the given surface, and which are coincident with the lines of curvature, be Asi and 2; of which let As2 lie along a generating line of the developable surface, and Asi be at right angles to it; then in the limit, if d fa be the angle subtended by dsx at the centre of principal curvature, dsi = pi dfa. Let the axis of the cylinder be parallel to the generating line of the developable at the given point, and from a point in the axis of the cylinder let normals be drawn to the cylinder parallel to normals drawn to the developable surface along A*i, and let Ao-j be the intercepted arc of the circle on the surface of the cylinder; then Ao"! = A fa; also let a line equal to As2 be taken on the surface of the cylinder and perpendicular to Ao-! through one of its extremities; by this process therefore

da = Pldfads2, rfa _ J_

di = dfa ds2, 'ds pi'

and therefore the curvature of a developable surface at a given point is equal to the curvature of the principal normal section which is perpendicular to a generating line.

I must in conclusion observe that a different definition of curvature of a surface has been given by Madame Sophie Germain; for which I must refer the reader to her own memoir on the Curvature of Surfaces; see Crelle, VII. 1; and Recherches sur la Theorie des Surfaces filastiques, Paris, 1821 and 1826.



418.] Many theorems, which occur in common Algebra and elsewhere, are true, not because the subjects of them have any special nature as quantities, but because they conform to certain laws and combine in certain manners; and these theorems therefore are equally true of all symbols which are subject to these laws. For the sake of an example let us take the binomial theorem, and its ordinary proof: » simple factors of the form x—a, x—b, ... are multiplied together; and the law of their combination having been detected, we are thence led to a general form; and if we assume a = b = c = ..., we obtain the general expansion of {x—a)n. Now this theorem is true, not because x, a, b, ... are symbols of quantity, but because being symbols of quantity they satisfy certain laws, in conformity with which the said theorem is. All other symbols therefore which satisfy these laws are subject to the same theorem. Similarly other algebraical theorems of expansion, and expansions such as those which have been given in the preceding pages for er, sin x, tan-1 x, and, in short, Maclaurin's and Taylor's Series, are true of all symbols which satisfy certain laws.

A distinction, and a division consequent upon that distinction, has been made of symbols; some are called symbols of operation, others symbols of quantity: those which indicate processes to be performed on subjects, such as symbols of differentiation; those of trigonometrical operations, as sin, tan, sec-1; those of logarithmic affection, as loga, c( >, &c. are called symbols of operation: whereas the subjects on which these operations are performed are called symbols of quantity. Thus if a, b, x, y are Bymbols of the ordinary quantities of Algebra, they are called symbols of quantities. The distinction however is scarcely accurate; for may not x be the same as x x 1? in which case x symbolizes a process, that of multiplication, performed on unity, and is a symbol of operation; but it is useful enough for our purpose.

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