two, it is formally more correct to regard them as infinite in number, but as so related that any two of them which are independent contain by implication all the rest.

Such considerations are easily extended to differential equations of the higher orders.

Geometrical illustrations.

11. Geometry, by its peculiar conceptions of direction, tangency, and curvature, all developed out of the primary conception of the limit, Art. 1, throws much light on the nature of differential equations.

As the simplest illustration let the equation of a straight line

be taken as the complete primitive, a and b being arbitrary

Of these equations, (4), which is free from arbitrary constants, is the general differential equation of the second order of a straight line; and (2) and (3), each of which contains one of the original arbitrary constants, are the two differential equations of the first order. Moreover, each of these differential equations expresses some general property of the straight line-(2), that its inclination to the axis is uniform; (3), that any intercept, parallel to the axis of y, between the

straight line and a parallel to it through the origin will be of constant length; (4), that a straight line is nowhere either convex or concave;-and this property, which does not involve, in the same definite manner as the others do, the considerations of distance and of angular magnitude, is evidently the most absolute of the three.

The equation of the circle is

(x − a)2 + (y — b)2 = p2.

..(5),

and if we regard a and b as arbitrary constants the corresponding differential equation of the second order will be

expressing the property that the radius of curvature is in

variable and equal to r.

which is the general differential equation of a circle free from arbitrary constants. And the geometrical property which this equation also expresses is the invariability of the radius of curvature, but the expression is of a more absolute character than that of the previous equation (6). For in that equation we may attribute to r a definite value, and then it ceases to be the differential equation of all circles, and pertains to that particular circle only whose radius is r. The equation (7) admits of no such limitation.

Monge has deduced the general differential equation of lines of the second order expressed by the algebraic equation ax2 + bxy + cy2+ ex+fy = 1.

But here our powers of geometrical interpretation fail, and results such as this can scarcely be otherwise useful than as a registry of integrable forms.

From the above examples it will be evident that the higher the order of the differential equation obtained by elimination of the determining constants from the equation of a curve, the higher and more absolute is the property which that differential equation expresses.

We reserve to a future Chapter the consideration of the genesis of partial differential equations as well as of ordinary differential equations involving more than two variables.

EXERCISES.

1. Distinguish the following differential equations according to species, order, and degree, and take account of any peculiarities dependent upon their coefficients.

2. Explain the term 'complete primitive,' and form the differential equations of the first order of which the following are the complete primitives, c being regarded as the arbitrary constant, viz.:

3. Form the differential equations of the second order of which the following are the complete primitives, c and c' being regarded as arbitrary constants.

4. State the criterion by which it may be determined whether differential equations are derived from a common primitive.

5. Shew that the differential equations

아

are not derived from a common primitive involving a and b as arbitrary constants.

6. Shew that each of the following pairs of equations, in

which

Р stands for is derived from a common primitive,

dy
dx'

and determine the primitive;

(2) y− xp=a (y2+p), and y − xp=b (1 + x2p). ✓

7. How many first, second, third, &c. integrals belong to the general differential equation of lines of the second order given in Art. 11, and how many of each order are independent?

8. From the equation (y—b)2= 4m (x − a) assumed as the primitive, deduce 1st the differential equations of the first order involving a and b as their respective arbitrary constants; 2dly the general functional expression for all differential equations of the first order derivable from the same primitive. ✔

9. Of what primitive involving two arbitrary constants would the functional equation

represent all possible differential equations of the first order?

10. How many independent differential equations of all orders are derivable from a given primitive involving x, y, and n arbitrary constants?