158 MOTIONS OF FREE PARTICLES. [CHAP. IX. 



Thus we cannot deduce from the law of gravitation an exact 

 account of the motions of the bodies forming the solar system. 

 But there are a number of circumstances which conduce to the 

 possibility of deducing from this law such an approximate account 

 of the motions in question as shall be sufficiently exact to agree 

 with observation over a long period of time. Among these we 

 may mention (1) that the mass of the Sun is great compared with 

 that of the other bodies, even the mass of Jupiter being less than 

 _^th part of that of the Sun, (2) that all the orbits are nearly 

 circular and lie nearly in one plane. 



Now it follows from the first of these statements that all 

 the forces acting on any planet are small compared with the 

 attraction of the Sun, and thus an approximate description of 

 the motion might be obtained by leaving these forces out of 

 account. The approximate equations can be completely solved, 

 as we have seen in Article 175. If then, starting at any instant, 

 we could conceive that a planet moved under no force except the 

 attraction of the Sun, it would describe an ellipse with the Sun 

 in one focus; and, since at starting it would have the position 

 and velocity which it actually has, this ellipse would touch the 

 actual path. 



The ellipse in question is known as the "instantaneous ellipse," 

 and the motion in it is of the kind described in Articles 53 55. 

 The method of Planetary Theory is to determine this ellipse and 

 to determine how it changes from time to time. For the deter- 

 mination of the ellipse we observe that the plane of the ellipse 

 will cut any other plane through the Sun in a line, so that, in 

 particular, the orbit of any planet cuts the Earth's orbit in a line 

 through the Sun, this line is known as the " line of nodes ; " the 

 position of the line of nodes and the angle between the two planes 

 determine the plane of the ellipse ; the angle in question is known 

 as the " inclination." The position of the ellipse in its plane is 

 determined by means of the angle contained between the line of 

 nodes and the axis major. The shape and size of the ellipse are 

 determined by its eccentricity and its major axis. One further 

 element is necessary in order to determine the position in terms 

 of the time, and this is arranged for by choosing an epoch from 

 which to measure time. 



