Um-1°

Hence Au must be a solution of A,P=0, and therefore a

function of u1, U 2, . . . . . .

And so for the others.

results therefore that the transformed system is

It

u, u,,..... um being the actual independent variables of the system.

But the transformation having involved no loss of generality, for a new system of m independent variables was simply substituted for an old one, the condition

(4;4; — A;A;) P=0,

satisfied before, will continue to be satisfied in the new system represented symbolically in the form

AP=0, A,P=0, ...... A„P=0.

Any common integrals of this system will also be common integrals of the previous system. For as functions of

И1, Ида
29...... m-1

they will satisfy the first equation of that system, and they will satisfy the other equations, because the present system is but a transformation of those. The converse is equally manifest.

Thus a system of n partial differential equations containing m independent variables and satisfying a certain condi

tion, has in virtue of that condition been converted into a system of n-1 equations between m -1 independent variables, and satisfying the same condition. This then is convertible into a similarly constituted system of n-2 equations containing m2 independent variables, and so on till we arrive at a final single partial differential equation containing m n + 1 independent variables. This equation has m—n, that is, r integrals, and these are the common integrals of the system (1).

But the system of ordinary differential equations corresponding to (1) is in number r, and is satisfied by all the common integrals of that system. Hence these differential equations must admit of reduction to the exact form.

7. We may deduce from the above investigation the following Rule.

To integrate a system of simultaneous linear partial differential equations of the first order.

RULE. Reduce the equations to the homogeneous form (1), express the result symbolically by

AP=0, AP=0, ...... A„P = 0,

and examine whether the condition

is identically satisfied for every pair of equations of the system. If it be so, the equations of the auxiliary system, Prop. I., will be reducible to the exact form, and their integrals being

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

the complete value of P will be F(u, v, w, ...), the form of F being arbitrary.

If the condition be not identically satisfied, its application will give rise to one or more new partial differential equations. Combine any one of these with the previous reduced

system, and again reduce in the same way. With the new reduced system proceed as before, and continue this method of reduction and derivation until either a system of partial differential equations arises between every two of which the above condition is identically satisfied, or, which is the only possible alternative, the system

appears. In the former case the system of ordinary equations corresponding to the final system of partial differential equations will admit of reduction to the exact form, and the general value of P will emerge from their integrals as above. In the latter case the given system can only be satisfied by supposing Pa constant.

Ultimately then the determination of P depends on the solution of a system of ordinary differential equations reducible to the exact form. This does not mean that each equation of the system is reducible to the exact form, but that the equations may be combined together so as to form an equal number of equivalent equations of the exact form. Generally when we know this combination to be possible it is easy to effect it, and best to endeavour to do so. We might however employ the method of the variation of parameters as follows. Supposing p the number of differential equations make all but p+1 of the variables constant, integrate the reduced system, and then seek to satisfy the unreduced system by the same series of integrals with the arbitrary constants as new variables. The successive integrations and transformations of this method would amount to the same thing_as those upon which the second part of the demonstration of Prop. III. rests*.

Lastly, given a system of ordinary differential equations containing a superfluous number of variables without knowing how many integrals they admit, we must, supposing Pc to be any integral, construct the corresponding system

It was thus indeed that the author was first led to that theory.'

of homogeneous partial differential equations satisfied by P, and apply to them the foregoing Rule.

8. Ex. Required the integrals of the simultaneous partial differential equations

Representing these in the form A‚P=0, A‚P=0, it will be found that the equation

(4,4, — 4,4,) P = 0

becomes, after rejecting an algebraic factor,

dP dP
X +
dz dt

=

0,

and the three equations prepared in the manner explained in the Rule will be found to be

No other equations are derivable from these. We conclude that there is but one final integral.

To obtain it, eliminate

dP dP dP

dx' dy' dt

and equate to 0 the coefficient of in the result. We

find

dz (t + 3x2) dx - ydy - xdt = 0,

the integral of which is

dP

dz

An arbitrary function of the first member of this equation is the general value of P.

[It appears from the manuscript that another example was to have been added here.]