of article 346, give values of x, y, z, in terms of the time which will determine the motion of the sun in space; for if the first of them be multiplied by S+m, and added to These equations give the motion of the sun in space accompanied by m; and as they are the same for each body, if Em be substituted for m, they will determine the absolute motion of the sun attended by the whole system, when the relative motions of m, m', m'', &c., are known. 367. But in order to ascertain the values of x, y, z, the equations (89) must be integrated. Since these equations are linear and of the second order, their integrals must contain six constant quantities. They are also symmetrical and so connected, that any one of the variable quantities x, y, z, depends on the other two. M. Pontécoulant has determined these integrals with great elegance and simplicity in the following manner. 368. If the first of the equations (89) of elliptical motion multiplied by y, be subtracted from the second multiplied by x, the result will be where c, c', c', are arbitrary constant quantities introduced by integration. Again, if the first of the same equations be multiplied by 2dr, the second by 2dy, and the third by 2dz, their sum will be 2dxd x+2dydy+2dzdz But whence dt2 r2 = x2 + y2 + 2o ; and the integral of the preceding equation is containing the seven arbitrary constant quantities c, c', c'', ƒ, f', ƒ", and a. 370. As two equations of condition exist among the constant quantities, they are reduced to five that are independent, consequently two of the seven integrals are included in the other five. For if the first of these equations be multiplied by z, the second by y, and the third by x, their sum is cz + c'y + c'x=0. (92) Again, if the fourth integral multiplied by c, be added to the fifth multiplied by c', but this coincides with the sixth integral, when f" = - fc + f'c', or f'"d' +ƒ'c' + fe = 0. C' fc The six arbitrary quantities being connected by this equation of condition, the sixth integral results from the five preceding. If the squares of f, f', and f", from the fourth, fifth, and sixth integrals be added, and f+f12 + ƒ112 = l2, but cz + c'y + c'x = 0; hence cdz + c'dy + c'dx = 0; and comparing this equation with the last of the integrals in article 369, it will appear that thus, the last integral is contained in the others; so that the seven integrals and the seven constant quantities are in reality only equal to five distinct integrals and five constant quantities. 371. Although these are insufficient to determine x, y, z, in functions of the time, they give the curve in which the body m moves. For the equation cz + dy + c''x = 0 is that of a plane passing through the origin of the co-ordinates, whose position depends on the constant quantities c, c', c". Thus the curve in which m moves is in one plane. Again, if the fourth of the integrals in article 269 be multiplied by z, the fifth by y, and the sixth by, their sum will be but in consequence of the three first integrals in article 369, it be the time; for if du represent the indefinitely small arc mb, fig. 73, contained between then Smr and Sb = r + dr, (mb)2 = dx2 + dy2 + dz2 = r2dv2 + dr2 ; but the sum of the squares of the three first of equations (91) is 373. Thus the area rdv described by the radius vector r or Sm is proportional to the time dt, consequently the finite area described in a finite time is proportional to the time. It is evident also, that the angular motion of m round S is in each point of the orbit, inversely as the square of the radius vector, and as very small intervals of time may be taken instead of the indefinitely small instants dt, without sensible error, the preceding equation will give the horary motion of the planets and comets in the different points of their orbits. Determination of the Elements of Elliptical Motion. 374. The elements of the orbit in which the body m moves depend on the constant quantities c, c', c', f, f', f", and " order to determine them, it must be observed that in the equations (89) the co-ordinates x, y, z) are SB, Bp, pm, fig. 74; but if they be referred to 7S the line of he equinoxes, so that SD Dpy', pmz', and if y SN ENP, the longitude of the node and inclination of the orbit on the x', fixed plane be represented by 0 fig. 74. D B . In a ዝበ and ; it is evident, from the method of changing the co-ordinates in article 225, that |
