310 A Short History of Astronomy [CH. XL 



of the Mecanique Celeste. These were generally accepted 

 in lieu of Burg's, which had been in their turn an im- 

 provement on Mason's and Mayer's. 



Later work on lunar theory may conveniently be regarded 

 as belonging to a new period of astronomy (chapter xin., 

 286). 



242. Observation had shewn the existence of inequali- 

 ties in the planetary and lunar motions which seemed to 

 belong to two different classes. On the one hand were 

 inequalities, such as most of those of the moon, which went 

 through their cycle of changes in a single revolution or a 

 few revolutions of the disturbing body ; and on the other 

 such inequalities as the secular acceleration of the moon's 

 mean motion or the motion of the earth's apses, in which 

 a continuous disturbance was observed always acting in the 

 same direction, and shewing no signs of going through a 

 periodic cycle of changes. 



The mathematical treatmer t of perturbations soon shewed 

 the desirability of adopting different methods of treatment 

 for two classes of inequalities, which corresponded roughly, 

 though not exactly, to those just mentioned, and to which 

 the names of periodic and secular gradually came to be 

 attached. The distinction plays a considerable part in 

 Euler's work ( 236), but it was Lagrange who first 

 recognised its full importance, particularly for planetary 

 theory, and who made a special study of secular inequalities. 



When the perturbations of one planet by another are 

 being studied, it becomes necessary to obtain a mathematical 

 expression for the disturbing force which the second planet 

 exerts. This expression depends in general both on the 

 elements of the two orbits, and on the positions of the 

 planets at the time considered. It can, however, be divided 

 up into two parts, one of which depends on the positions of the 

 planets (as well as on the elements), while the other depends 

 only on the elements of the two orbits, and is independent of 

 the positions in their paths which the planets may happen 

 to be occupying at the time. Since the positions of planets 

 in their orbits change rapidly, the former part of the 

 disturbing force changes rapidly, and produces in general, 

 at short intervals of time, effects in opposite directions, first, 

 for example, accelerating and then retarding the motion of 



