Application of this method to the motion of comets, supposing them to be acted upon only by the attraction of the sun: it gives, by an equation of the seventh degree, [784], the distance of the comet from the earth. The mere inspection of three observations, made within very short intervals of each other, will suffice to discover whether the comet is nearer to the sun, or farther from it, than the earth is, [780'*] § 31
Method of finding, as accurately as is necessary, by means of three observations, the geocentric longitudes and latitudes of a comet, and their first and second differentials, divided by the corresponding powers of the element of the time, [787'] § 32
Determination of the elements of the orbit of a comet, when we know, at a given instant, its distance from the earth, and the first differential of this distance divided by the element of the time. Simple method of allowing for the excentricity of the orbit of the earth, [788 — 800]. § 33
When the orbit is a parabola, the greater axis becomes infinite, and this condition furnishes another equation, of the sixth degree, [805], to determine the distance of the comet from the earth. § 34
Hence we may obtain various methods of computing a parabolic orbit. Investigation of the method from which we may expect the most accurate result, and the greatest simplicity in the calculation, [806—811] § 35, 36
This method is divided into two parts : in the first, is given a method of approximation, to find the perihelion distance of the comet, and the time of passing the perihelion, [811" — 820] ; in the second, is given a method of correcting these two elements, by three distant observations, and then deducing from these the other elements, [820'" — 832] § 37
Accurate determination of the orbit, when the comet has been observed in both of its nodes, [833—841] §38
Method of finding the ellipticity of the orbit, when the ellipsis is very excentric, [842 — 849]. § 39
CHAPTER V. GENERAL METHODS OF FINDING THE MOTIONS OF THE HEAVENLY BODIES, BY SUCCESSIVE APPROXIMATIONS 475
Investigation of the alterations which must be made in the integrals of differential equations, to obtain the integrals of the same equations increased by certain terms, [850 — 859]. . . § 40
Hence we derive a simple method of obtaining the rigorous integrals of linear differential equations, when we know how to integrate the same equations deprived of their last terms, [861 — 871"]. § 41
We also obtain an easy method of computing the integrals of differential equations, by successive approximations, [872 — 875] § 42
Method of eliminating the arcs of a circle, which occur in these approximate integrals, when they do not really exist in the rigorous integrals, [876 — 892] § 43
Method of approximation, founded on the variations of the arbitrary constant quantities, [897-912] §45
CHAPTER VI. SECOND APPROXIMATION OP THE CELESTIAL MOTIONS ; OR THEORY OF THEIR PERTURBATIONS 504
Formulas of the motions in longitude and latitude, and of the radius vector in the disturbed orbit. Very simple form under which they appear, when only the first power of the disturbing forces is noticed, [913—932] § 46
