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which is the condition that the parameter a must satisfy in order that the spiral whose equation is r = că may be its own evolute.

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which is also the equation to an epicycloid.

CHAPTER XIII.

APPLICATIONS OF THE DIFFERENTIAL CALCULUS TO GEOMETRY

OF THREE DIMENSIONS.

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de

IF

F il, y, ) = 0 be the equation to a curved surface, the equation to the tangent plane at a point x, y, z is

dF

dF
(it' - x) + (y - y) + (x'- ) -- = 0,
dx

dy where x', y', z' are the current co-ordinates of the tangent plane, x, y, z those of the point of contact.

If the equation to the surface consist of a function homogeneous of n dimensions in x, y, c equated to a constant, the equation to the tangent plane becomes

[ , d E , dF , d .
it dx + y dy** da

- + = =- = NC,
F (x, y, z) = c being the equation to the surface.

If p be the perpendicular from the origin on the tangent plane,

dF dF dF.

- tz

dy dx 12 dF ?

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and if the function be homogeneous of n dimensions,

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The equations to a normal at a point x, y, s are

a' 4-y x' -
dF

d F dF:

dx - dy do
Ex. (1) The equation to the Ellipsoid being

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The perpendicular on the tangent plane from the origin is given by the equation

If we wish to find the locus of the intersection of the tangent plane with the perpendicular on it from the centre, we have to combine the equation to the tangent plane,

with the equations of a line perpendicular to it, and passing through the origin

a2 c bêy cha

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Multiplying each term of the equation to the tangent plane by the corresponding member in these last expressions, x, y, % are eliminated, and we have for the locus of the intersections

y"? + y2 + x2 = (a.z."? + b2 y'? + c'z'?).

This is the equation to the surface of elasticity in the wave Theory of Light. (2) Let the equation to the surface be

xyz = m*.
The equation to the tangent plane is

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The intercepts on the tangents are

2. = 3x, y'o = 3y, s = 37, and the volume of the pyramid included between the tangent

b plane and the co-ordinate planes is 24

. 9.xyz 9a3

22 The volume of this pyramid is smaller than that of any other pyramid formed with the co-ordinate planes by a plane passing through the point x, y, z.

The length of the perpendicular from the origin is given by

1 1/1 1 1 1
p 3 lir

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e sin

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(5) The equation of the hélicoide développable is

% (x2 + y2 – a?)? sint-lot-02} + y cos **** - 10+ lh

1} = a. a The cosine of the angle which the tangent plane makes with the plane of my is

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dF 27,

dx = (2x2 + e* – a?)]. From these expressions the cosine of the inclination of the tangent plane to the plane of my is found to be

27 a

(1° + 4 *a*)3 The inclination is therefore constant, and equal to that of the helix, which is the directrix of the surface.

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