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 P=c 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. the

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

+ = 0, dz dt

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.

dP

dz

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

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.]

CHAPTER XXVI.

HOMOGENEOUS SYSTEMS OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS.

1. THE theory of homogeneous systems of linear partial differential equations in which when expressed in the symbolic form

is for all combinations represented by i and j satisfied in virtue of the constitution of the symbols A, A,, forms the subject of important researches by Jacobi (Nova Methodus... Crelle's Journal, Vol. LX. p. 1). The following are the most important of his results.

1st. An integral of any one equation of the system being found, other integrals of the same system may be obtained without integration, by a process of derivation founded upon the condition (2).

Let be an integral of the first equation of the system. Then is the equation

identically satisfied.

Δφ= 0

Also the condition (2) being satisfied in virtue of the constitution of the symbols, we have

and in particular, making i=1, and separating the terms,

Δ.Δ,Φ - Δ,Δ,φ = 0,

which reduces by a prior equation to

Δ.Δ,φ = 0.

It appears from this that A,, if it do not reduce to a constant, is an integral of the first equation 40, and, if it prove to be not a mere function of p, a new integral.

This process may be repeated upon the new integral with a similar alternation of results. It will be evident from this that if we confine our attention to the two equations

Δ.Ρ=0, Δ.Ρ= 0,

and suppose, as before, o to be an integral of the first, then will

Δ.Φ, Δ. (ΔΦ), Δ.{Δ.(Δ.Φ)},...

2

or, as these may be expressed,

Δ.Φ. Δ. Φ, Δ. Φ, ...

be also integrals of the first equation; and this process of derivation may be continued until we arrive at an integral A," which is not independent, but is expressible as a function of prior integrals

Δ.Φ, Δ. Φ....... Δ. -φ,

and, sooner or later, such a result must present itself, since the number of independent integrals is finite.

It is further seen that the most general symbolic form of an integral derivable from the root integral & is

A.A.......A",

a, B, ...... μ, being positive integers.

The above remarkable theorem was in some degree anticipated by the researches of Poisson.