containing the seven arbitrary constant quantities c, c', c", f, f', f", 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',

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

they give

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

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 + c'y + d'x = 0

is that of a plane passing through the origin of the co-ordinates, whose position depends on the constant quantities c, d, 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

ƒz+f'y+ƒ"x+ H{.2® +y3+z2) = c(zdr−xdz)+(ydz—zdy) μ(x®

dt

dt

r

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 de

pend on the constant quantities c, c', c', f, f', ƒ"', 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 S the line of he equinoxes, so that SD = x', Dpy', pm = z', and if SN ENP, the longitude of the node and inclination of the orbit on the fixed plane be represented by

. In

a

fig. 74.

P

D

B

N

and ; it is evident, from the method of changing the co-ordinates

in article 225, that

Thus the position of the nodes and the inclination of the orbit are given in terms of the constant quantities c, d, c".

375. Now = x2 + y2 + z2, and rdr = xdx + ydy + zdz, but at the perihelion the radius vector r is a minimum; hence dr =0, therefore xác + ydy + zdz = 0.

Let x, y, z,, be the co-ordinates of the planet when in perihelio, then, substituting the values of c, c', c", from 269 in the equations in f' and f" of the same number, and dividing the one by the other, the result in consequence of the preceding relation will be

But if , be the angle SE, the projection of the longitude of the

which determines the position of the greater axis of the conic section.

by means of the last of the integrals (91) the result will be

but at the extremities of the greater axis dr = 0, because the radius vector is either a maximum or minimum at these points, therefore at the aphelion and perihelion

The sum of these two values of r is the major axis of the conic section, and their difference is FS or double the eccentricity.

376. Thus a is half of AP, fig. 75, the major axis of the orbit, or

it is the mean distance of m from S; and

h2

1- is the ratio

μα

of the eccentricity to half the major axis. Let this ratio be repre

Thus all the elements that determine the nature of the conic section and its position in space are known.

377. The three equations

r2=x2+y2+z2, μr—h2+fz+f'y+f"z = 0, and c''x+c'y+cz = 0, give x, y, z, in functions of r; but in order to have values of these co-ordinates in terms of the time, r must be found in terms of the same, which requires another integration. Resume the equation

To integrate this equation, a value of r must be found from