36. The singular integral of the differential equation of the second order, above referred to, has been found to be 16 (1+x2) y - 8x3y, -16xy,+x-16y,=0. Ex. 2, Art. 10.

Shew that this singular integral has for its complete primitive

(16y+ 4x2 + x1)* = x (1 + x2)* − log {(1 + x2)* − x} +h,

h being an arbitrary constant-and that this is a singular solution of the proposed differential equation of the second order.

37. The same singular integral has for its singular solution 16y+4x2+x=0. Prove this. Have we a right to expect. that this will satisfy the differential equation of the second order?

38. By reasoning similar to that of Chap. VIII. Art. 14, shew that a singular integral of a differential equation of the form yn+f(x, y, y1... Yn-1)=0 will render the integrating factor of that equation infinite.

39. Differential equations of the form

dzy dy

dx2

can be integrated by obtaining two first integrals of the respective forms x=f(p, c), y=f1 (p, c), and equating the values of p.

40. Prove the assertion in Art. 9, that a singular solution of a singular integral of a differential equation of the second order is in general no solution at all of the equation given.

CHAPTER XI.

GEOMETRICAL APPLICATIONS.

1. IN what manner differential equations afford the appropriate expressions of those properties of curves which involve the ideas of direction, tangency or curvature, has been explained in Chap. I. Art. 11.

Of the suggested problem in which from the expression of a property involving some one or more of the above elements it is required to determine, by the solution of a differential equation, the family of curves to which that property belongs, some illustrations have also been given in the foregoing Chapters.

Here we propose to consider that problem somewhat more generally.

The following expressions furnished by the Differential Calculus are convenient for reference.

For a plane curve referred to rectangular co-ordinates x and

y, representing also

dy

d'y

by P, dx2

by q,

dx

Intercept on axis y = y — xp.

Dist. from origin to foot of normal = x + yp.

Perpendicular from (a, b) on tangent=

Perpendicular from (a, b) on normal =

y − b − (x − a) p
(1+p2)

Co-ordinates (a, B) of centre of curvature

To these may be added the well-known formulæ for the differentials of arcs, areas, &c.

It is evident from the above forms that problems which relate only to direction or tangency, give rise to differential equations of the first order-problems which involve the conception of curvature to equations of the second order.

When the conditions of a geometrical problem have been expressed by a differential equation, and that equation has been solved, it will still be necessary to determine the species of the solution-general, particular, or singular, as also its geometrical significance.

2. The class of problems which first presents itself, is that in which it is required to determine a family of curves by the condition that some one of the elements whose expressions are given above shall be constant.

Ex. 1. Required to determine the curves whose subnormal is constant.

Here y dx dy

=

= a, and integrating,

The property is seen to belong to the parabola whose parameter is double of the constant distance in question, and whose axis coincides with the axis of x, while the position of the vertex on that axis is arbitrary.

Ex. 2. Required a curve in which the perpendicular from the origin upon the tangent is constant and equal to a. Here we have

y− xp=a (1+p3),

an equation of Clairaut's form, of which the complete primitive is

The former denotes a family of straight lines whose distance from the origin is equal to a, the latter a circle whose centre is at the origin, and whose radius is equal to a. And here, as was noted generally by Lagrange, the singular solution seems to be, in relation to geometry, the more important of the two.

3. A more general class of problems is that in which it is required to determine the curves in which some one of the foregoing elements, Art. 1, is equal to a given function of the abscissa x.

Ex. 1. Required the class of curves in which the subtangent is equal to ƒ (x).

Thus if the proposed function were x, we should have

Ex. 2. Required the family of curves in which the radius of curvature is equal to f(x).

in which it only remains to substitute for X its value, and effect the remaining integration.

If f(x) is constant and equal to a, we find

and this represents a circle whose centre is arbitrary in position, and whose radius is a.

A yet more general class of problems is that in which it is required that one of the elements expressed in Art. 1 should be expressed by a given function of x and y.

An example of this class is given in Chap. VII. Art. 10.

4. We proceed in the next place to consider certain problems in which more than one of the elements expressed in Art. 1, are involved.

B. D. E.

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