d

1

or

3.

dm am

а

log {(am — a') (x + my) + cm − c'} = C.

[The next Article seems to have been intended to appear in the enlarged form of Chap. II.; but I cannot discover what precise position it would have occupied. I conjecture that the above demonstration" refers to Chap. II. Arts. 2, 3; and I have accordingly supplied a reference to equation (3) of Chap. II.

I had myself drawn Professor Boole's attention to Chap. II. Arts. 2, 3. The geometrical process of Chap. 11. Art. 3, appears to have been first given by D'Alembert in his Opuscules, Vol. iv. p. 255. D'Alembert calls it a demonstration; it seems to me only an illustration, at least in the brief form of the text and that such was Cauchy's opinion may perhaps be inferred from the elaborate investigation given by Moigno, to which Professor Boole refers in Art. 5 of the present Chapter.

I had also drawn Professor Boole's attention to the statement at the end of Chap. 11. Art. 12, that only one arbitrary constant was involved. Accordingly Article 5 of the present Chapter developes this statement, and Article 4 seems intended to bear on the same subject.]

4. In the above demonstration the relation between y and x is regarded as one of pure magnitude, and the interpretation of the differential equation becomes a limiting case of that of the equation of finite differences (Eq. (3), Chap. II.). But if we represent x and y by the rectangular co-ordinates

of a moving point on a plane the differential equation may be interpreted directly. For supposing it reduced to the form

we see that the direction of motion is constantly assigned as a function of the co-ordinates of position. The entire motion is therefore determinate as soon as the initial point is fixed. The result of the motion is a line or curve wholly continuous or subject to irregularities according to the nature of the function f(x, y). That the arbitrariness of origin is geometrically equivalent to the appearance of a single arbitrary constant in the relation connecting x and y may be shewn thus. y = (x, y, x)

Let

be the relation between x and y indicated by the supposed motion, x, y, being the initial point of departure. Then this point being on the line of motion, x, y, are particular values of х and y, so that we have from the above equation

which establishes a relation between x and y., and shews that there exists virtually but one arbitrary constant.

5. It is proved in Art. 3, Chap. II., that the constants x, y, initial values of the variables x, y in the solution of the differential equation of the first order, are necessarily equivalent to one arbitrary constant. I shall shew from the form of the above solution that this a priori condition is actually satisfied.

Developing the expression for y [see Eq. (30) of Chap. 11.] in ascending powers of x, we have

the summation extending from n=r to no. Forming

hence the differential coefficients of A, with respect to x。 and Yo, and reducing by (28), we shall find

Eliminate between these equations f(x, y), and we have

Therefore, by Prop. I., A, is a function of A,, so that the solution reduced to the form (32) contains but the single arbitrary constant Д..

It remains to notice that the solution must be applied only under the conditions of convergency, i.e. under the condition that the ratio of the nth to the (n - 1)th term tends to a limit less than unity as n tends to infinity. For a discussion of the failing cases of this test see Finite Differences,' Chap. v. Generally it is desirable, in order to secure rapid convergency, to divide the interval x-x into separate equal portions, to each of which the general theorem of solution may be applied. If x-x be very small the theorem may be approximately represented by

y — Y。 = ƒ (x., Y.) (x − x).

On these principles Cauchy has founded remarkable methods of solution, which deserve attention from the commentary on the limits of error on their application by which they are accompanied (Moigno, Vol. II. pp. 385-434).

(7)

CHAPTER XX.

ADDITIONS TO CHAPTER VII.

1. [THIS Article relates to Art. 2 of Chap. VII.]

We

The sense in which (9) may be said to constitute the general solution of the differential equation is this. obtain from it

giving any particular value to C this will geometrically represent a curve consisting of two branches, and giving to C'every possible value we obtain an infinite system of such curves, each consisting of two branches. The aggregate of branches thus obtained is evidently the same as the aggregate of curves given by the two primitives (5) and (6), unrestricted by any connexion between C1 and C2. In this sense then the solution (9) is general, that it includes all the particular relations between y and x which are deducible from the original primitives (5) and (6). And it is only in this sense not general that it groups these relations together in a particular manner.

To the expression of the complete primitive a certain variety of form may be given without affecting its generality in the sense above affirmed. Thus, if to the solutions of the component differential equations we give the forms

-ax

ye-c1 = 0, logy + ax-c2 = 0,

we should have, by the same procedure, as the expression of the complete primitive,

an equation which may equally with (9) be regarded as the complete primitive of the differential equation given, and which in geometry represents the same totality of branches of curves as (9), with this difference only, that they are differently paired together.

2. [This Article relates to Art. 3 of Chap. VII.]

The question will here naturally arise, Since if V = c be a solution of one of the component differential equations, f(V) = c, in which ƒ (V) denotes any function of V, is also a solution, by Chap. IV. Art. 3, why not give to the complete primitive the form

{ƒ. (V.) — c} {ƒ, (V2) — c} {ƒn (Vn) — c} = 0, or the stricter form

.......

ƒ1 (V1) ƒ1⁄2 (V2). · .ƒn (Vn) = 0 ..................................(F'), in which fi(V), f. (V),... fa (Va), denote arbitrary functions of V1, V2,..., V, respectively-stricter because the presence of arbitrary constants and functions in the previous form is a superfluous generality? It is replied that though the form just given is analytically more general than (15), it is not more general than (15) with such freedom as is permitted in the interpretation of the arbitrary constants. In a physical or geometrical application we should not only be permitted to assign a particular value to the arbitrary constant in (15), so deducing what in reference to its source would then be termed a particular primitive, but to combine the results of different determinations of c together, so as to obtain every form of solution which is implied either in the functional equation (F), or in its component primitives

The same considerations justify us in speaking of (15) as the complete primitive, and not as a complete primitive.