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[The next Article may be considered supplementary to Chap. xiv. Art. 10.]

Singular Solutions of partial Differential Equations.

8. Legendre's theory developed in Chap. XXIII. for ordinary, may be applied also without essential change to partial, differential equations. Regarding the independent variable z as receiving an infinitesimal change dz through infinitesimal change, not in the values of the independent variables

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but in the values of the arbitrary constants of the complete or in the forms of the arbitrary functions of the general integral, and performing upon the given equation the operation denoted by 8, we shall obtain a linear partial differential equation for determining the general value of dz corresponding to any particular given value of z. If that linear equation be of a lower order than the differential equation given, then the equation expressing the value of z+ 8z will be a limiting form of a solution less complete or less general than the complete or general solution of the differential equation given, and the given solution, formed by making the infinitesimal constants in the limiting form actually 0, will be singular.

Conversely, to deduce singular solutions without the knowledge of the complete or the general integral, we ought to construct the equations of condition for the reduction of the equation determining oz to a lower order than the equation given, and the most general solution of the differential equations of condition so formed, will be the most general expression for the singular solutions of the differential equation given. :

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Representing the first member of the equation by F, we
have, on operating by d,
dF doz dF doz dF

dp dx * dą dy * dz



and the conditions


0, - 0,

dp dq
necessary to reduce the equation for 8z to a lower order give

(px 2y) q 2mx* = 0, (px 2y) (px - 3qy) = 0.

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From these we find

p= 3m*x*yt, q=m*xyt,

p definite and simultaneous values of p and q, which being substituted in the given equation lead to


9 as those

and this, as it gives the same values of


and obtained before, will necessarily satisfy the given equation. It is therefore a solution, and from the nature of the analysis, a singular one.

Legendre shews that this singular solution is also deducible from the general integral of the given partial differential equation. That integral is the result of the elimination of a from the two equations {$ (a)}' – 2axo (a) + az – mxy = 0,

| , {$ (a) – ax} $ (a) 2x® (a) +z = 0. To deduce the singular solution he supposes $ (a) to be not simply a function of a, but a function of a and of one or both of the independent variables. He expresses the varia

= ax.

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

{24 (a) - 2ax} 8(a) = 0; therefore

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

2mbæty? as before.

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 S.

[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 Transactions.

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.]




1. The term simultaneous is here applied to a system of partial differential equations, to siguify 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

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