INTRODUCTION 



The reciprocal gravitation of matter produces disturbances in the motions of the 

 heavenly bodies, causing them to deviate from the elliptic paths which they would 

 follow, if they were attracted only by the sun. The determination of the amount 

 by which the actual place of a planet deviates from its true elliptic place at any 

 time is called the problem of planetary perturbation. The analytical solution of 

 this problem has disclosed to mathematicians the fact that the inequalities in the 

 motions of the heavenly bodies are produced in two distinct ways. The first is a 

 direct disturbance in the elliptic motion of the body; and the second is produced 

 by reason of a variation of the elements of its elliptic motion. The elements of the 

 elliptic motion of a planet are six in number, viz: the mean motion of the planet 

 and its mean distance from the sun, the eccentricity and inclination of its orbit, 

 and the longitude of the node and perihelion. The first two are invariable- the 

 other four are subject to both periodic and secular variations. 



The inequalities in the planetary motions which are produced by the direct 

 action of the planets on each other, and depend for their amount only on their 

 distances and mutual configurations, are called periodic inequalities, because they 

 pass through a complete cycle of values in a comparatively short period of time ; 

 while those depending on the variation of the elements of the elliptic motion are 

 produced with extreme slowness, require an immense number of ages for their 

 full development, and are called secidar inequalities. The general theory of all the 

 planetary inequalities was completely developed by La Grange and La Place nearly 

 a century ago; and the particular theory of each planet for the periodic inequali- 

 ties was given by La Place in the Mecanique Celeste. 



The determination of the periodic inequalities of the planets has hitherto 

 received more attention from astronomers than has been bestowed upon the secular 

 inequalities. This is owing in part to the immediate requirements of astronomy, 

 and also in part to the less intricate nature of the problem. It is true that an 

 approximate knowledge of the secular inequalities is necessary in the treatment of 

 the periodic inequalities ; but since the secular inequalities are produced with such 

 extreme slowness, most astronomers have been content with the supposition that 

 they are developed uniformly with the time. This supposition is sufficiently near 

 the truth to be admissible in most astronomical investigations during the compara- 

 tively short period of time over which astronomical observations or human history 

 extends ; but since the values of these variations are derived from the equations 



( vii ) 



