(6) Let the surface be Fresnel's surface of elasticity, the equation to which is

a2x2 + b2y2 + c2 x2 = (x2 + y2 + ≈2)3.

The equation to the tangent plane is

(2 r2 − a2) x x' + (2 r2 − b2) y y' + (2 r2 − c2) xx′ = r1,

where 22 + y2 + x2. p2 =

The perpendicular from the centre on the tangent plane is

(x2 + y2 + x2)2

(a1x2 + b1y2 + c1z2) § °

When a curved line in space is given by the equations of two of its projections,

the equations to a tangent at the point x, y, ≈ are

The equation to the normal plane is

(x' − x) dx + (y' − y) dy + (≈′ – ≈) d z = 0).

The equation to the osculating plane is

(x' − x) (dy ď3 z − dz d2 y) + (y' − y) (dzď2 x − dx ď z)

(7)

Let the given curve be the helix, the equations to which are

In finding the equation to the osculating plane we may for simplicity assume dx = 0, that is, make ≈ the independent variable. This assumption readily gives us as the equation to the osculating plane,

h (xy' - yx') + a2 (≈′ −‚≈) = 0.

In both of these equations if we make = 0, y' = 0, we find ; that is, both planes cut the axis of ≈ at the same point, which is the corresponding co-ordinate of the point in the curve.

(8) Let a curve of double curvature be formed by the intersection of two cylinders, the axes of which cut each other at right angles.

The equations to the curve are

x2 + x2 = a2, y° + x2 = b°,

the point of intersection of the axes of the cylinders being taken as origin, and the axes as the axes of r and y. The equations to the tangent are

The equation to the osculating plane is, making ≈ the independent variable, and therefore dx = 0,

b2x23 x' — a2y3y + (a2 − b3) ≈3 x' = a2b2 (a2 — b3).

When a curved line in space is not given by the equations to its projections, but by the equations to any two surfaces,

F (x, y, x) = 0, F1 (x, y, x) = 0,

of x, y, z: and these values are then to be substituted in the equations to the tangent, and to the normal and osculating planes.

(9) Let the curve be that formed by the intersection of a sphere and an ellipsoid. It is determined by the equations

=

=

a2 (b2 - c2)

*

b2 (c2 - a) c (a - b)

The equation to the normal plane is

This curve is the spherical ellipse; that is, it is a curve described on the surface of a sphere such that the sum of the arcs of great circles drawn from any point in the curve to two fixed points on the surface of the sphere is constant.

(10) Let the curve of double curvature be the equable spherical spiral. This is formed by the intersection of a sphere with a right cylinder the radius of whose base is one half of that of the sphere, and which passes through the centre of the sphere. The equations to the curve are therefore

x2 + y2 + x2 = 4r2, y2 + x2 = 2rx,

the axis of

being taken parallel to the axis of the cylinder, and the axis of a passing through the centre of the base of the cylinder. The equations to the tangent are

y (y' − y) = (r − x) (x′ − x), ≈ (≈′ - x) = r (x' − x).

SECT. 2. Curvature.

If a curved surface be given by an equation of the form

the greatest and least radii of curvature of the normal sections passing through a point x, y, ≈ are given by the equation p2 (rt - s2) - pk { (1 + q2) r − 2pqs + (1 + p2) t } + k2 = 0, where is the radius of curvature.

If the surface be given by an equation of the form

the equation for determining the radii of maximum and minimum curvature is

-U2u'2-V2v'2-W2w'2+2VWv'w'+2WUw'u'+2UVu'v'=0.

This equation is much longer than the preceding, but from its symmetry it is more useful in practice, and is very frequently much simplified by the vanishing of some of the quantities which it contains*.

(1) Let the given surface be the Ellipsoid,

where p is the perpendicular from the centre on the tangent plane. Hence the equation becomes

From the last term of this it appears that the product of the greatest and least radii of curvature of normal sections is constant for all points for which the perpendicular on the tangent plane is constant.

(2) Let the surface be the paraboloid, the equation to which is

For a demonstration of this equation see Cambridge Mathematical Journal, Vol. I. p. 137. The reader is referred also to Gregory's Solid Geometry, where the same equation is presented under a more simple and elegant form.