This is the simplest case of the "curves of pursuit," and the problem may be expressed thus: A point P moves along a straight line, and is pursued by a point Q, whose velocity is to that of P always in the ratio of m to n, find the path of Q, supposing the line joining the points at the beginning of the motion not to coincide with the direction of the motion of P.

Bouguer, Mémoires de l'Académie, 1732.

(22) Find the curve in which the radius of curvature is equal to the normal.

Now the radius of curvature and the normal may lie on the same or on different sides of the curve: this will be indicated by taking the radius of curvature with a negative or a positive sign. Hence by the condition,

(23) Find the curve which has an evolute similar to itself.

This remarkable problem if attempted by means of reference to rectilinear co-ordinates would be quite impracticable. Euler*, however, has by a most ingenious method reduced the problem to a very simple shape. Instead of using rectilinear or polar co-ordinates he refers the curve to its radius of curvature, and the angle which that line makes with a line passing through the first point of the curve to be investigated. There is thus nothing left arbitrary except the first point in the curve.

Let

be the radius of curvature, the angle which it makes with the line passing through the first point of the curve. Then if s be the arc of the curve measured from the same point,

x = fd&p cos p, y = fdopsin ;

and x and y are thus known in terms of the co-ordinates which we are to employ.

Now let AS (fig. 61) be the curve, A'S' its evolute which is to be similar to AS, and let A"S" be the evolute of A'S', and therefore similar to it and to AS: let PP', QQ' be the radii of curvature at the successive points P, Q. Then considering the small arcs PQ and P'Q' as coinciding with the arcs of the circles of curvature, we see that the elemental sectors PQ'Q and P'Q'Q' are similar, and therefore

=

But by the property of the evolute ds' dp; therefore

In order that the evolute may be similar to the curve, we must have

p = ap,

a being the coefficient of similarity.

Hence for determining the curve we have the equation

This being a linear equation is easily integrated, and the result is

Substituting this value of p in the expression for a and y, a = Cfdpe" cos p, y = Cfdpe" sin p.

Whence

αφ

(y + B) (a + p) = (x + A) (ap − 1).

Or putting x and y for a + A and y + B, which does not

affect p

=

dy

dx

this becomes

a (rdy – ydx) = d + dụ,

which is the differential equation to the logarithmic spiral. That curve therefore is the only one which has an evolute similar to itself.

Euler in the memoir referred to above has considered the question much more generally, for he investigates the nature of the curve which has its nth evolute similar to itself, as well as the curve which has an evolute similar to itself but placed in an inverse position. This last is reduced to the previous case, for if the evolute be similar to the original curve but in an inverted position, the second evolute will

also be similar to the original curve and in the same position, and its radii of curvature will diminish while those of the first evolute increase, as will be seen in (fig. 62). It is easy to see that this condition is expressed symbolically by affecting the coefficient of similarity with a negative sign. The general equation for a curve which has its nth evolute directly similar to itself is

(24) Let us investigate a particular case of this last problem when n = 1 and a2 = 1, which implies that the evolute is equal to the curve but in an inverted position.

The integral of which is

p = C cos + C, sin = C cos (p + a).

Since the angle a depends only a depends only on the line from which is measured we may make it equal to zero, so that