5. The above theory may be extended to all systems which are composed of n differential equations of the first order and degree connecting n+1 variables. Assumex (independent) and 1, 2, ..., (dependent) as the variables of the system. Then there exist n differential equations of the form Pdx+P1dx,+P2dx2... + F ̧dx=0.........................(10), 2 n P, P,, &c. being functions of the variables. These equations exactly suffice to determine the ratios of the differentials dæ, dx,... dx, and thus assume the symmetrical form X, X, &c. being determinate functions of the variables. This premised, the solution of the system (11) depends upon the solution of a single differential equation of the nth order connecting two of the variables. Let us select for the two x and X1 dx X Differentiate the first of these n 1 times in succession, regarding x as independent variable and continually substitut ing for da dx 2 n dx ... dx their values as given by the n-1 last equations of the above system. We thus obtain, including the equation operated upon, n equations connecting with the primitive variables and therefore enabling us, 1st, to express the above n differential coefficients in terms of those variables, 2ndly, by elimination of the n-1 variables, ∞, ∞, ..., to deduce a single equation of the form Now this being a differential equation of the nth order, there exist, Chap. IX. Art. 1, n first integrals involving n distinct arbitrary constants and capable of expression in the form 1(x, In-1 d2x 1 _n-1 dx, d2x, d-1x, their ... If in this system we substitute for dx' dx2 dx"-1 values in terms of the primitive variables above referred to, we shall obtain a system of n equations of the form This is the primitive system sought. And thus the following Propositions are established, viz. 1st, that a system of differential equations of the first order connecting n+1 variables is expressible in the symmetrical form (11). 2ndly, that its complete solution depends on that of an ordinary differential equation of the nth order (13). 3rdly, that that solution consists of n equations connecting the primitive variables with n arbitrary constants and theoretically expressible in the form (15). These very important propositions were first established by Lagrange, but the above demonstration of them is taken from a memoir by Jacobi *(1827). It is not necessary, as is evident from the examples already given, actually to determine the n first integrals of the differential equation (13). The complete primitive and the successive equations obtained from it by differentiation enable us to ac * Ueber die Integration der partiellen Differential-Gleichungen erster Ordnung. Crelle, Tom. 11, p. 317. Werke, Vol. IX (1886), pp. 1-15, sup. 4-5. to complish the same object. Neither is it always necessary proceed to differential equations of an order higher than the first. This point will be illustrated in the following sections. Linear equations of the first order with constant coefficients. 6. The characters here mentioned have reference only to the dependent variables which are the true unknown quantities of the system. Thus the equation would be described as linear and with constant coefficients. The solution of any system of n such equations is by the foregoing general method reducible to that of an ordinary linear differential equation of the nth order with constant coefficients. And this method is in the two following respects the best of all, viz. 1st, because of its fundamental character, 2ndly, because it leads directly to the expression of the values of the dependent variables. The solution of such a system may however also be effected by the method of indeterminate multipliers, and this we propose here to exemplify. Its advantage is that it generally presents the equations of the solution under a common type, so that their discovery is made to depend upon the discovery of a single general form. Forsyth, p.2%, Ex. Given 2x.31 Multiplying the second equation by an indeterminate quantity m, and adding to the first, we have provided that we determine m so as to satisfy the condition In this equation it only remains to substitute in succession the two values of m furnished by (b). The two resulting equations, in which the arbitrary constants must of course be supposed different, will express the complete solution of the problem. When the values of m are equal, the form (c) furnishes directly only a single equation of the complete solution. We may deduce the other equation, either by the method of limits (assuming the law of continuity), or by eliminating x from the given system by means of (c), and then forming a new differential equation between y and t. It seems preferable however to employ the general method of Art. 5, by which all difficulties connected with the presence of equal or imaginary roots are referred to the corresponding cases of ordinary differential equations. 7. Simultaneous equations are so often presented under the symmetrical form (11) that the appropriate mode of treatment deserves to be carefully studied, especially as it possesses the superiority, always in point of elegance, and frequently in point of convenience, over other processes. It is known that each member of a system of equal fractions is equal to the fraction which would be formed by dividing any linear homogeneous function of their numerators by the same function of their denominators. Hence if we have a system of equations of the form in which we suppose t the independent variable, and Ta function of t only, then we shall have = dt_dx, ... n .(17). Hence, should the first member be an exact differential, the inquiry is suggested whether the multipliers m,... r cannot be so determined, whether as functions of the variables or as constants, as to render the second member such also. Now when the system of equations is linear and with constant coefficients this can always be effected. It may be observed that the character of the system is as manifest from inspection of the symmetrical form (16) as of the ordinary form. If the system be linear and with constant coefficients the denominators X,, X,, ... X, will, when considered with respect to the dependent variables ∞, ∞, ..., be linear and with constant coefficients. n In the employment of this method it is often of great advantage to introduce a new independent variable, and to consider all the variables of the given system as dependent upon it. We are thus enabled to secure the condition above adverted to, of having one member of the symmetrical system an exact differential. 36 Let us introduce a new variable t so as to give to the system |