386 Royal Astronomical Society, 



spending change in the equation of the centre, and cannot possibly 

 exceed that amount. But if the force have been for a long time 

 gradually altering the major axis, and consequently the periodic 

 time in the moon's orbit, then during the whole of that time the 

 moon has been performing her revolutions quicker or slovi^er than we 

 expected, and therefore at the end of that time she is in advance 

 or in retard of her expected place by an amount equal to the accu- 

 mulation of all the advances or retards in all the revolutions through 

 which the change has been going on. The planetary inequalities of 

 long period are all of this kind. The major axis here plays the same 

 part as the pendulum of a clock. If a small force acting for a year 

 pushed the seconds-hand forwards by an inch, the clock would be 

 merely a few seconds wrong ; but if in the same time it shortened 

 the pendulum by an inch, the clock would have gained fifty hours ; 

 and if the time occupied by the change had been greater, the dis- 

 turbance in the clock indication would have been proportionably 

 greater. 



In order then to find inequalities of long period in the motion of 

 the moon produced by Venus, we must seek for some alternate in- 

 crease and decrease, occupying a very long period, in the force by 

 which Venus draws the moon from the earth. 



No such slow increase and decrease have been found in the general 

 force by which Venus disturbs the moon. 



The next point of inquiry is, whether a combination of the changes 

 in the force of Venus with the changes in the position of the moon 

 in its orbit can produce a force, which, for a very long time together, 

 gradually increases the force drawing the moon from the earth, and 

 then for an equal time gradually diminishes that force. 



A force which acts in opposite ways, nearly on opposite sides of 

 the moon's orbit (pulling the moon from the earth on one side and 

 pushing it towards the earth on the other side), may produce this 

 effect, provided the period of the change in the nature of the force 

 (from pulling to pushing) correspond nearly, but not exactly, with 

 the time in which the moon moves from apogee to perigee. For 

 (as we have seen) the effect of a certain force of Venus is to produce 

 a greater disturbing force on the moon at apogee than at perigee ; 

 and this force, or a change in this force, will, at apogee, produce a 

 greater effect on the dimensions of the moon's orbit than at perigee, 

 both because the disturbing force is actually greater, and because it 

 acts on the moon when the moon's velocity is smaller. Therefore, 

 if a pulling force, gradually increasing in magnitude, act on the 

 moon at apogee, it will gradually increase the dimensions of the 

 moon's orbit : if a corresponding pushing force act at perigee, it 

 will gradually diminish the dimensions of the moon's orbit; but the 

 former prevails, and the orbit will gradually increase in size. If 

 after a time the pulling force at apogee gradually diminish, and at 

 length become a pushing force, while the pushing force at perigee 

 gradually diminishes, and at length becomes a pulling force, then 

 the orbit will gradually diminish in size. And this change of forces 

 would be produced by such a modification in Venus's force, as that 



