SECT. III. DISTURBING FORCES. 15 



the sun and planet, and is called the radial force : it produces 

 periodical changes in the distance of the planet from the sun, 

 and affects the form and position of the orbit in its own plane. 

 The third, which may be called the perpendicular force, acts at 

 right angles to the plane of the orbit, occasions the periodic 

 inequalities in the planet's latitude, and affects the position of 

 the orbit with regard to the plane of the ecliptic. 



It has been observed, that the radius vector of a planet, 

 moving in a perfectly elliptical orbit, passes over equal spaces 

 or areas in equal times ; a circumstance which is independent of 

 the law of the force, and would be the same whether it varied 

 inversely as the square of the distance, or not, provided only 

 that it be directed to the centre of the sun. Hence the tangential 

 force, not being directed to the centre, occasions an unequable 

 description of areas, or, what is the same thing, it disturbs the 

 motion of the planet in longitude. The tangential force some- 

 times accelerates the planet's motion, sometimes retards it, and 

 occasionally has no effect at all. Were the orbits of both planets 

 circular, a complete compensation would take place at each revo- 

 lution of the two planets, because the arcs in which the accelera- 

 tions and retardations take place would be symmetrical on each 

 side of the disturbing force. For it is clear, that if the motion 

 be accelerated through a certain space, and then retarded through 

 as much, the motion at the end of the time will be the same as 

 if no change had taken place. But, as the orbits of the planets 

 are ellipses, this symmetry does not hold : for, as the planet 

 moves unequably in its orbit, it is in some positions more 

 directly, and for a longer time, under the influence of the dis- 

 turbing force than in others. And, although multitudes of 

 variations do compensate each other in short periods, there are 

 others, depending on peculiar relations among the periodic times 

 of the planets, which do not compensate each other till after one, 

 or even till after many revolutions of both bodies. A periodical 

 inequality of this kind in the motions of Jupiter and Saturn has 

 a period of no less than 918 years. 



The radial force, or that part of the disturbing force which 

 acts in the direction of the line joining the centres of the sun 

 and disturbed planet, has no effect on the areas, but is the cause 

 of periodical changes of small extent in the distance of the planet 

 from the sun. It has already been shown, that the force pro- 



