and as dx, dy, dz are arbitrary, though infinitesimal, this equation can be true only when Hereby we have criteria whether (37) is or is not an exact differential, for since any) - (OT), (09) - ( m), and certain (open) )() = (ant), on e) = (ar), (41) if (40) are true, we have which equations are called the conditions of integrability of (37); and if they are fulfilled we can integrate as follows: Since pdx is the x-partial differential of the general integral, the x-integral of Pdx will give us the whole function of x which enters into the general integral; similarly the y-integral of Qdy will give the whole function of y; and the z-integral of rdz, the whole function of z: if therefore we add to the x-integral of Pdx those functions of y which are in the y-integral of Qdy and are not in the x-integral of pdr; and if again we add to the sum those functions of z which are in the z-integral of r dz, and which have not already entered, the result will evidently contain all the variable part of the general integral, and therefore by the addition of an arbitrary constant the general integral will be obtained. Hence we have where x, y, ... are severally functions of x, y, ... only; and are determined in the manner explained above. In this case it is evident that if the variables are separated, the criteria of integrability are satisfied. Also that if pdx +qdy + Rdz can be divided into portions of which some are at once perceived to be exact differentials, it is sufficient to ascertain by the criteria whether the other parts are also exact differentials. Ex. 1. yz dx + zxdy + xy dz = 0. and the conditions of integrability are satisfied. Let ur, U,, u, denote the several partial integrals; then, taking indefinite integrals, 1 = (pdx = yzdx = xyz ; similarly, u, = xyz ; w: = xyz; .. u = xyz — k3 = 0; where k is an arbitrary constant; and this is the general integral. , xdx + y dy+zdz , zdx — x dz. 42 +3 ax? dx + 2 by dy+cdz=0. (x2 + y2 +za) 22 +22 The several parts of this equation are so evidently exact differentials, that it is unnecessary to apply the criteria ; and for the general integral, if k is an arbitrary constant, we have u = (x2 + y2 + z2)+ tan . + ax: +by2 + cz+k. 374.] When the criteria of integrability are satisfied, the general integral may be expressed in terms of definite partial integrals as follows: as the process is similar to that of Art. 372, it is unnecessary to repeat every step of it; let the differential equation be fi(x, y, z)dx +f2(x, y, z) dy +fz(x, y, z) dz = 0; .:. u = 1 f1(x, y, z)dx+v, (43) where v is a function of y and %; the number of which conditions is the same as that of n things taken two and two together, that is, is ; and when these are satisfied, and the n partial-integrals are found, the general integral may be determined from them by a process similar to that employed in the cases of two and three independent variables. We may also express the general integral in terms of definite partial integrals in the following manner. Let the coefficients of the differentials in (46) be fı (X1, X2, ... Xn), fz (x1, X2, ... X»), ...fr (X1, X2, ... Xn); and let the inferior limits of integration be X, X,, ... X, : then by a process similar to that of the last Article, u = 1"} (x1,x2,..X»)dx, + [ "$($1,g..w)dx, +.. SECTION 3.— Integration of homogeneous equations of two variables. 377.] Differential equations of the first order and degree can generally be integrated only when they satisfy the criteria of integrability; and therefore when an equation does not fulfil these conditions, our first object is to investigate, if it be possible, some mode of so transforming it that its equivalent may be in the required form : the principal means which are useful for such a transformation are (1) an introduction of new variables by way of substitution, (2) the multiplication of the equation by a factor which will render it an exact differential, and which is commonly called an integrating factor : these processes we go on to examine. First, let us take the case of two variables x and y, and suppose the equation to be Pdx + Q dy = 0; (51) let us suppose that the criterion of integrability is not satisfied, but that p and q are homogeneous functions of x and y of n dimensions : then dividing through by m", so that 20" may stand as a common factor, (51) takes the form 21 f (%) dx + 1" + (%) dy = 0. (52) PRICE, VOL. II. 3Y Let y = xz; therefore dy = x dz +z dx; and neglecting the factor 2", (52) becomes f(x) dx + (2) {xd2 +z dx} = 0; (53) .ã * f(x) +20(z); in which equation the variables are separated; and consequently the condition of integrability is satisfied; and thus the integration depends on that of two simple differentials of one variable. Instead of arranging the equation (51) in the form (52), wherein x" is the common factor, it might equally as well have been put into the form |