individual member of another set. If some, even if one,

of one

set be different from all of another set this is sufficient to establish them as different sets, and to entitle them to the use of distinctive differential and integral symbols of operation.

6. In fixing the values of a new set of independent variables in terms of the original set, it is of importance to take care that each member of the new set be really an independent variable; i.e. that it be such as cannot be expressed in terms of the other members of its own set. Unless this condition of independence be observed, we shall be working under an erroneous hypothesis.

7. Under the authority of Art. 5 we are at liberty to assume, whenever it can be done usefully, that one of the variables of each new set shall be identical with one of the variables of the original set,-for example, we may assume Tt, in which case equation (1) of Art. 4 becomes

Dudu+du.D1x+du. D1y+............ (1).

But when we use this equation, which will always be known from the occurrence of the symbol D or St, we must remember that there remain only (n-1) new independent variables (the quasi-constants) of the set to be determined, one of them (t) being already fixed upon in using this equation.

8. In determining what shall be the forms of the constituent members of a new set of variables, we are at liberty to introduce n arbitrary hypotheses if we use equation (1) of Art. 4; but only (n−1) arbitrary hypotheses if we employ equation (1) of the last Article. The arbitrariness of these hypotheses is however overruled by the necessity that the resulting members of a set of variables determined from them must be independent (Art. 6).

(5)

CHAPTER II.

DETERMINATION OF QUASI-CONSTANTS.

FIRST ORDER.

EQUATIONS OF THE

Ex. 1. To integrate du + adu= bu.

Let us change the independent variables to t, §. Then the formula of Art. 7 becomes for this case, which is one of two independent variables,

Dudu+Dx.du....... (1).

We are at liberty to make one hypothesis (Art. 8) for the determination of the value of §; let us assume D1 = a. We make this choice, because by this assumption the right-hand member of the formula (1) will be rendered identical with the left-hand member of the proposed equation; so that we shall have Dubu;

.. Da, and Dubu,

and these two equations, taken simultaneously, are together exactly equivalent to the proposed equation. Integrating them we have (see Art. 2)

x = at + §, and etu= F(§).

The former gives the value of §; and this value being written

in the latter, we have the following as a relation that must always exist between u and the independent variables t, x, u=ebt F(x-at).

This is therefore the required integral.

9. REMARK. According to Art. (2) we ought to have written x = at +ƒ(§) instead of x = at +, to complete the integration of D=a: but had we done so the final integral would not have been affected thereby and we were moreover at liberty to take either x- at, or any function of x- at as the value of §. Under this license no additional generality would be gained by the use of ƒ§ instead of §. We shall always make use of this liberty by adding either §, or a function of έ, after one of the integrations in any proposed problem. Ex. 2. To integrate du + adu+bd1u=cu.

In this case, which is one of three independent variables, we have to change the variables to t, §, n; and the formula of Art. 7 becomes

Dudu + Dix.d ̧u+ D1y.du.

We are here at liberty to make two hypotheses for the determination of the quasi-constants § and 7. We therefore assume Dixa, Dy=b; which gives Du = cu. These three

Ꭰ :

=

equations, when taken simultaneously, are equivalent to the proposed equation.

We shall employ the first and second for fixing the values of & and n;

.. x = at + §, y=bt+n, and eu = F' (§, n).

And by substituting in the last of these the values of and found from the other two we find, as the required integral,

u = et F(x-at, y-bt).

Comparing this, which is a case of two independent variables, with Du= du+ D ̧x.d ̧u, we find the two following: equations which when taken simultaneously are equivalent to the proposed equation;

The former by integration gives x2+, which fixesthe value of : and the two equations combined give

a2x

Ex. 4. To integrate du+4. du=a2oæ.

t

Ex. 5. To integrate du+%.d ̧u+&.du=”u

α

пи

variables, with

t

Comparing this, which is a case of three independent

Dudu+Dx.d ̧u+ D¿y.du,

t

t

Ex. 6. To integrate da V. du-d¿V. d ̧u = 0.
Divide by dV, and compare with

The latter gives by integration u = §, and the former may be written in the form 0=dV+ D. dV which is = D,V. And hence, by integration,

V = F(§) = F (u).

Hence, in an equation of the form proposed, the dependent variables u, V are necessarily functions of one another. We may write the integral under the form

f(u, v) = 0.

х

y2

Ex. 7. To integrate du – 2. du + 4. du = YZ

y t