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


= f (x, y),

da 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 (, y). That the arbitrariness of origin is geometrically equivalent to the appearance of a single arbitrary constant in the relation connecting w and y may be shewn thus. Let

· y = $(xo, Yo, 2C) be the relation between x and y indicated by the supposed motion, 2o, y, being the initial point of departure. Then this point being on the line of motion, Xo, y, are particular values of w and y, so that we have from the above equation

y.= (2., Yo, xo), 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 xo, Yo, 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

A2x2 A..2 y= A.+ 4,2+

+ &c., ........ (32) 1.2

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the summation extending from n=r to n=co. Forming hence the differential coefficients of A, with respect to a

and Yo, and reducing by (28), we shall find


+fi (2o, y) da.

dA. dy.

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

dx, dy. dy, dx,

. 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 A..

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- %. = f (2., y.) (x – 20.). 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).

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y = Cear;

1. [This Article relates to Art. 2 of Chap. VI..]

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

y = Ce-at; 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 c, and c. 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

ye* C = 0, logy + ax - C,= 0, we should have, by the same procedure, as the expression of the complete primitive,

(yeas — c) (log y + ax – c) = 0,

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.

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

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

{fi(V) – c} {f. (V.) – c}.........{fn ( Vn) – c} = 0, or the stricter form

fi(V) f2(V) (Vm) = 0 ............(F"), in which fi(V), f:(V.),... fu( Vn) denote arbitrary functions

, ( of V1, V2, ..., Vn 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

Vi=C, V = C, ...,

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





1. [The Singular Solutions of Differential Equations of the First Order received great attention from Professor Boole, and the Chapter devoted to that subject is one of the most valuable and important in his work. He continued his researches after the publication of his first edition, and intended to reconstruct the Chapter with great improvements in the second edition. After carefully examining the manuscripts I came to the conclusion that it would be very difficult to rewrite this portion of the work so as to connect the old matter with the new; and thus it seemed best to reprint the original Chapter viii. with corrections of obvious misprints, and to print the matter intended for the revised form in the present volume. The plan gives rise to some repetition; but this seems unimportant, compared with the advantage of preserving in the author's own language all that he left on an interesting and important point which he had carefully studied.

2. It may be of service to the student to reproduce the substance of some remarks on his Chapter viii. which were sent to Professor Boole soon after the publication of his first edition; for there is evidence in his manuscripts that he paid great attention to such remarks while engaged in the revision of his work, and thus the reason and the meaning of some of his additions and changes may be made more obvious. These remarks will occupy the next Article.

3. The two pages beginning with “ And these conditions are sufficient,” and ending with “do not lead to conflicting

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