which is therefore the differential equation to the evolute.

(9) The equation to the logarithmic curve is y = ac";

is the equation to the evolute.

In curves referred to polar co-ordinates the most convenient mode of finding the equation to the evolute is by the relation between Ρ and r.

If p and r be the co-ordinates of the curve,

P and r,

be the radius of curvature; then p = f(r) being the equation to the curve,

Between these four equations we can eliminate p, r, p,

and so find a relation between P, to the evolute.

(10) Let p2 = r2 — a2.

and r which is the equation

Then pp, r2 = r2 + p2 − 2 p

= r2 − p2 = a2,

and p2 = r2 – p2 = a2.

د,

Hence p, and r, being both constants, the evolute is a

circle.

(11) In the logarithmic spiral p = mr,

the equation to a similar logarithmic spiral.

The logarithmic spiral may even be its own evolute; that is, one convolution of the curve may be the evolute of another convolution. To find the condition that this should be the case, let

be the equation to the curve. Let P (fig. 52) be a point in the curve, PN the normal at that point touching a point Q in the convolution which is the evolute of the convolution AP. Then since the curve makes a constant angle with its radius vector, the angle SPT must be equal to the angle SQP; that is, PSQ must be a right angle. Hence the radius SQ is separated from the radius SP by some whole number of circumferences together with three right angles, or if

But Q being a point in the evolute, r = ar,, so that

which is the condition that the parameter a must satisfy in

0

order that the spiral whose equation is rea may be its own evolute.

Substituting for c22 its value in terms of p,,

r2

and p =

c2 - a2

which is also the equation to an epicycloid.

be the equation to a curved surface, the equation to the tangent plane at a point x, y, z is

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

If the equation to the surface consist of a function

homogeneous of n dimensions in a, y,

equated to a con

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

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

and if the function be homogeneous of n dimensions,

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

Ex. (1) The equation to the Ellipsoid being

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

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

x'2 + y'2 + x22 = (a2 x2 + b2 y2 + c2x22)§.