when y1, Y2, .... Y1 are the roots of the algebraic equation

Y2

provided that for the first equation a= eo, and for the second a= ε°.

If n = 2, the above equations assume the forms

6. [The two principal papers by Mr Harley on the differential equations exhibited on page 190 are the following:

(1) On the Theory of the Transcendental Solution of Algebraic Equations, Quarterly Journal of Mathematics, Vol. v. pages 337...360.

(2) On a certain class of Linear Differential Equations. Manchester Memoirs. Third Series. Vol. II. pages 232...245.

In a letter bearing date January 13, 1864, Professor Boole pointed out to Mr Harley that his second equation might also be deduced from the general theorem discussed in Art. 5. Employing the above notation the deduction may be presented in the following form.

The equation (11) will reduce to

and for the transformation of (III) we have

These substitutions being effected we arrive, after some slight reductions, at the following equation,

n” [(n − 1) D′ — m]” ̄1 D'u' — (n − 1) [nD′ — m − 1]”eou' = 0,

which, making m=1 and u'=t, gives

n” [(n − 1) D' — 1]”~1 D't — (n − 1) [nD′ — 2]′′eot = 0,

an equation which admits of reduction. In fact, operating on both members with (D − 1), and determining the constant, as in the former case, by the aid of the Lagrangean expansion, we find

n"-1 [(n − 1) D']"1t — (nD' — n − 1) [nD' ~ 2]"2ot = [n-1]” ̄1o,

which is Mr Harley's second equation.

The references and deduction here given were to have been added to the memoir which is cited in page 189, according to Professor Boole's desire; but by some accident

they were not printed, and the omission was not discovered until after his death.

Mr Harley has lately succeeded in obtaining the following extension of Professor Boole's theorem.

is satisfied by the mth power of any root of the equation

From this he deduces the following; the differential equa

tion

is satisfied by the mth power of any root of the equation

For the materials of this Article I am indebted to Mr Harley.]

CHAPTER XXXI.

THE JACOBIAN THEORY OF THE LAST MULTIPLIER.

1. A SYSTEM of n differential equations of the first order and degree containing n+1 variables admits of n integrals of the form

...

И19 И 29 un being independent functions of the original variables. When n— When n-1 of these integrals have been found they enable us to eliminate n − 1 variables, with their differentials, from the given system of equations, and so to obtain a single final differential equation of the first order between the two remaining variables. The final equation admits of being made integrable by a factor, and its solution so found would constitute the nth and last integral of the system. We propose in this Chapter to develope the theory of the above integrating factor as established by Jacobi. The term 'principle of the last multiplier,' which is more usually employed, seems objectionable; for the essence of Jacobi's discovery consisted not in demonstrating the existence or the nature of the last integrating factor, but in the peculiar form of the method which he gave for its determination, and in the relations which are implied in that form. The discovery may be briefly said to consist in this; viz. that instead of forming by means of the n-1 known integrals the final differential equation between two variables and applying methods analogous to those of Chap. V., to determine its integrating factor, we construct antecedently to all integration a linear partial differential equation of the first order, any one integral of which

will enable us to assign an integrating factor of the final differential equation, whatever the order of the previous integrations may have been. Again, this partial differential equation depending for its construction only upon the form of the system given, we can often by examining it affirm beforehand that if all the integrals but one of the system be in any way found, the final integral will be deducible by quadratures. This happens in the case of the most important of all systems of differential equations-that of Dynamics.

Further, an ordinary differential equation of the nth order being reducible to a system of n differential equations of the first order, Jacobi's theory may here also enable us to predicate the possibility of the last integration when the previous integrations have been effected.

Beginning with a single differential equation of the first order reduced to the form

in which X and Y are functions of the two variables x and y, we know by Chap. v. that the integrating factor μ will be given by the solution of the partial differential equation

the form of which should be carefully noticed.

Consider next a system of two differential equations of the first order expressed in the general form

X, Y, and Z being functions of the three variables x, y, z, and suppose one integral, represented by

(x, y, z) = c

(3),

to be known. The function

(x, y, z), or, as we shall express