(5) The equation of the hélicoide développable is

The cosine of the angle which the tangent plane makes with the plane of xy is

Now ( cos- y sin 0) = x2cos20+ y2 sin2 0 -2xy sin cos 0, and from the equation to the surface

2xy sin cos 0 = a2 - a2 sin2 0 y cos 0; therefore

From these expressions the cosine of the inclination of the tangent plane to the plane of xy is found to be

Σπα

(h2 + 4π2 a2)}'

The inclination is therefore constant, and equal to that of the helix, which is the directrix of the surface.

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

a2x2 + b2y2 + c2x2 = (x2 + y2 + x2)2.

The equation to the tangent plane is

(2 r2 − a3) x x' + (2 r2 − b2) yy' + (2r2 − c2) xx′ = r2,

where r2 = ∞2 + y2 + x2.

The perpendicular from the centre on the tangent plane is

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, z are

The equation to the normal plane is

(x′ − x) dx + (y' − y) dy + (≈' — ≈) dx = 0.

The equation to the osculating plane is

(x' — x) (dy ď2 ≈ – dz d2 y) + (y′ − y) (dzďx − dæď 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 d'x = 0, that is, make ≈ the independent variable. This assumption readily gives us as the equation to the osculating plane,

h (xy' − yx') + a2 (≈′ − x) = 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, y2 + x2 = b2,

the point of intersection of the axes of the cylinders being taken as origin, and the axes as the axes of

The equations to the tangent are

xx′ + xx′ = a2, yy′ + zx′ = b2.

The equation to the normal plane is

and y.

The equation to the osculating plane is, making ≈ the independent variable, and therefore d2x = 0,

b2x23 x' — a2y3y + (a2 − b2) ≈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, z) = 0, F1 (x, y, z) = 0,

of x, y, 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

x2 y2 22

+

a2 b2 c

From these we find

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).

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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 + q°) r − 2pq8 + (1 + p2) t} + k1 = 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

P

P

- 2u' VW (u - 2) - 20'UW (v) - 2 w UV (w - ?)

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