tion of (a) derived from this new source by 8, and operating on the first equation with 8, finds

therefore

{24 (a) — 2ax} 84 (a) = 0 ;

$(a) = ax.

Substituting this in the equations of the general integral, and eliminating a, we find

as before.

2= = 2m3x3y3

Legendre states his theory of the derivation of the singular solutions of partial differential equations from the equations themselves with great brevity, but still as a general theory. And there is nothing in the statement that carries with it any apparent restriction upon either the order or the degree of the equations given. Until however we are in possession of a perfect theory of the genesis of partial differential equations we shall not be entitled to say that Legendre's theory of their singular solutions is a perfect one; for until then we cannot even define, in a perfectly general way, the nature of the operation denoted by 8.

[The next three Chapters all relate to the subject of partial differential equations of the first order. The manuscripts do not appear to have received their final revision from Professor Boole. It is certain that he intended the contents of Chapter XXV. to form a part of the new edition; and it is highly probable, although not certain, that the contents of Chapter XXVI. and Chapter XXVII. were also to be included.

The three Chapters are mainly derived from two memoirs by Professor Boole, published in the Philosophical Trans

actions.

The first memoir is entitled On Simultaneous Differential Equations of the First Order in which the Number of the Variables exceeds by more than one the Number of the Equations: it occupies pages 437...454 of the Philosophical Transactions for 1862.

The second memoir is entitled On the Differential Equations of Dynamics. A sequel to a Paper on Simultaneous Differential Equations: it occupies pages 485...501 of the Philosophical Transactions for 1863.

The first memoir was finished before Professor Boole had seen Jacobi's researches, which are cited at the beginning of Chapter XXVI; these researches indeed could only just have been published. In his second memoir Professor Boole describes Jacobi's methods, refers to his own already published, and points out the nature of the connexion between them.]

CHAPTER XXV.

ON SYSTEMS OF SIMULTANEOUS LINEAR PARTIAL DIFFERENTIAL EQUATIONS OF THE FIRST ORDER, AND ON ASSOCIATED SYSTEMS OF ORDINARY DIFFERENTIAL EQUATIONS.

1. THE term simultaneous is here applied to a system of partial differential equations, to signify that in that system there is but one dependent variable, the general expression of which, as a function of the independent variables satisfying all the equations at once, is the object of search. All linear partial differential equations of the first order being reducible to the homogeneous form, we shall presuppose this reduction here. Under this form indeed the problem actually presents itself in Geometry, in the theory of partial differential equations of the second order, and in Theoretical Dynamics.

We are sometimes led, in connexion with the same class of inquiries, to systems of ordinary differential equations marked by the peculiarity that the number of the variables exceeds by more than one the number of the equations. Such systems are intimately connected with the former-stand to them indeed in a similar relation to that which the Lagrangean auxiliary system bears to the single partial differential equation from which it arises. The theory which explains this connexion, and grounds upon it the method of solution of both systems will form the subject of the present Chapter.

Connexion of the Systems.

2. PROP. I. The solution of a system of simultaneous linear partial differential equations of the first order may be

made to depend upon that of a system of ordinary differential equations of the first order in which the number of the variables exceeds by more than one the number of the equations.

The system of partial differential equations being reduced to the homogeneous form, Chap. XXIV. Art. 6, let n be the number of the equations, x,, x, x the independent Xn+r variables, and P the dependent variable.

Then from the n given equations determining

we obtain an equivalent system of equations which, by transposition of its terms to one side, assumes the reduced form

a single partial differential equation which, on account of the arbitrariness of A, A, ...... A, is equivalent to the system from which it was formed.

These equations being included in the previous system (3), any integrals

u = a, v=b, w=c, &c.

of them will be integrals of it. Therefore u, v, w,... will be values of P satisfying the partial differential equation (2). For they will be the only values which can satisfy it independently of A, A,,....... Hence they will satisfy the equivalent system (1), and the general integral of that system will be

F(u, v, w, ...) = 0

the form of F being arbitrary.

(5),

Thus the relation of the system (4) to the system (1) is the same as the relation of the auxiliary system of a single linear