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xviii as the cubes of the transverse axes of the orbits. This law extends to the motion of the satellites about their primary planets, [544"'^] §16

Second method of integration of the differential equations of the preceding article, [545—558]. § 17

Third method of integration of the same equations ; this method has the advantage of giving the arbitrary constant quantities in functions of the co-ordinates and of their first differentials, [559-597] §18,19

Finite equations of the elliptical motion ; expressions of the mean anomaly, of the radius vector, and of the true anomaly, in functions of the excentric anomaly, [606] § 20

General method of reducing functions into series ; theorems which result from it, [607 — 651]. §21

Application of these theorems to the elliptical motion. Expressions of the excentric anomaly, [657], the true anomaly, [668], and the radius vector of the planets, [659], in converging series of sines and cosines of the mean anomaly. Expressions in converging series, of the longitude, [675], of the latitude, [679], and of the projection of the radius vector, [680], upon a fixed plane but little inclined to that of the orbit § 22

Converging expressions of the radius vector, [683], and of the time, [690], in functions of the true anomaly, in a very excentric orbit. If the orbit be parabolic, the equation between the time and the true anomaly will be an equation of the third degree, [693], which may be resolved by means of the table of the motions of comets. Correction to be made in the true anomaly calculated for the parabola, to obtain the true anomaly corresponding to the same time, in a very excentric ellipsis, [695] § 23

Theory of the hyperbolic motion, [702] §24

Determination of the ratio of the masses of the planets accompanied by satellites, to that of the sun, [709] § 25

CHAPTIIR IV. DETERMINATION OP THE ELEMENTS OP THE ELLIPTICAL MOTION 393

Formulas which give these elements, when the circumstances of the primitive motion are known, [712 — 716']. Expression of the velocity, independent of the excentricity of the orbit, [720]. In the parabola the velocity is inversely proportional to the square root of the radius vector, [720"] §26

Investigation of the relation which exists between the transverse axis of the orbit, the chord of the described arch, the time employed in describing it, and the sum of the extreme radii vectores, [748, 750] § 27

The most convenient method of obtaining by observation the elements of the orbit of a comet, [753", &c.] §28

Formulas for computing, from any number of observations, taken near to each other, the geocentric longitude and latitude of a comet, at any intermediate time, with the first and second differentials of the longitudes and latitudes, [754, &c.] § 29

General method of deducing, from the differential equations of the motion of a system of bodies, the elements of their orbits, supposing the apparent longitudes and latitudes of these bodies, and the first and second differentials of these quantities, to be known, at a given instant, [760, &c.] § 30