Ex. 1. To determine the curves in which the radius of curvature is equal to the normal.

If the radius of curvature have the same direction as the normal we shall have

The first side multiplied by dx is an exact differential and

the equation of a circle whose centre is on the axis of x. If the direction of the radius of curvature be opposite to that of the normal, it will be necessary to change the sign of the first member of (1). Instead of (2) we shall have

and this equation not containing x, we may depress it to the

whence, x = c' + c log {y + (y2 — c2)$}

This equation, reduced to the exponential form

(5).

(6),

The solution therefore indicates a circle when the direction of the radius of curvature and of the normal are the same, but a catenary when they are opposed. The latter curve has, however, many properties analogous to those of the circle. (Lacroix, Tom. II. p. 459.)

Ex. 2. To find a curve in which the area, as expressed by the formula fydx, is in a constant ratio to the corresponding arc.

which, agreeing in form with the last differential equation of the preceding problem, shews that (5) represents the curve required, and connects together the properties noticed in the last two examples.

Ex. 3. Required the class of curves in which the length of the normal is a given function of the distance of its foot from the origin.

The differential equation is

y (1+p2)1 = ƒ (x+yp)

(1),

and it belongs to the remarkable class discussed in Chap. VII. Art. 9, where the complete primitive is given, viz.

y2 + (x − a)2 = {ƒ (a)}...........

....

(2).

This represents a circle whose centre is situated on the axis of at a distance a from the origin, and whose radius is equal to

f(a). It is evident that this circle satisfies the geometrical conditions of the problem.

But there is also a singular solution, found by eliminating the constant a between (2) and the equation derived from (2) by differentiation with respect to a, viz.

x−a+ƒ(a) ƒ' (a) = 0.

.......

·(3).

For instance, if ƒ (a) = n1 as we have to eliminate a between the equations

the equation of a parabola. While in this example the complete primitive represents circles only, the singular solution represents an infinite variety of distinct curves, each originating in a distinct form of the function f (a). Other illustrations of this remark will be met with.

The above problem was first discussed by Leibnitz, who did not, however, regard its solution as dependent upon that of a differential equation, but, establishing by independent considerations the equation (2), which constitutes in the above mode of treatment the complete primitive of a differential equation, arrived at a result equivalent to its singular solution by that kind of reasoning which is employed in the geometrical theory of envelopes. Indeed it was in the discussion of this problem that the foundations of that theory were laid (Lagrange, Calcul des Fonctions, p. 268).

5. A certain historic interest belongs also to the two following problems, famous in the earlier days of the Calculus, viz 'the problem of Trajectories' and the problem of 'Curves of pursuit.' These we shall consider next. They will serve to illustrate in some degree the modes of consideration by which the differential equations of a problem are formed when a mere table of analytical expressions suffices no longer,

Trajectories.

Supposing a system of curves to be described, the different members differing only through the different values given to an arbitrary constant in their common equation—a curve which intersects them all at a constant angle is called a trajectory, and when the angle is right, an orthogonal trajectory. To determine the orthogonal trajectory of a system of curves represented by the equation

Now representing this value by m, and the corresponding

dy

value of for the trajectory by m', we have, by the condition

dx

of perpendicularity, m' =—=1. Hence for the trajectory

or

m

which must be true for all values of c. Hence the differential equation of the orthogonal trajectory will be found by eliminating c between (1) and (2).

Were the equation of the system of intersected curves presented in the form

(x, y, a, b) = 0,

a and b being connected by a condition

† (a, b) = 0,

we should have to eliminate a and b between the above two equations, and the equation

Ex. 1. Required the orthogonal trajectory of the system. of curves represented by the equation y = cx".

Here y-cx", whence by (2)

=

the equation required. We see that the trajectory will be an ellipse for all positive values of n except n=1,—an ellipse, therefore, when the intersected curves are a system of common parabolas. The trajectory is a circle if n = 1, the intersected system then being one of straight lines passing through the origin. The trajectory is an hyperbola if n is negative.

Ex. 2. Required the orthogonal trajectory of a system of confocal ellipses.

The general equation of such a system is

2

x2 y2
+ = 1,
a2 b2

a and b being connected by the condition

a2 — b2 = h2,

where h is the semi-distance of the foci, and does not vary from curve to curve. Hence we have to eliminate a and b from the above equations, and the equation