SS* lioyal Astronomical Society. 



M. Hansen states that he has examined several inequalities of 

 long period in the moon's motion which hitherto have escajjed notice, 

 but that in no other instance does the coefficient amount to 1". 



In concluding the account of this remarkable discovery, it is gra- 

 tifying to add that it explains almost precisely the observed inequa- 

 lity in the moon's mean motion, which for the last fifty years has 

 troubled physical astronomers. 



After the reading by the Secretary of a portion of this paper, the 

 Astronomer Royal gave an oral explanation of its general subject in 

 the following manner : — 



The disturbing effect of Venus upon the moon is not the whole 

 attraction of Venus upon the moon, but the difference of the two 

 attractions, of Venus upon the moon and of Venus upon the earth. 

 Thus, when the moon is between the earth and Venus, the attrac- 

 tion of Venus upon the moon is stronger than that of Venus upon 

 the earth, and therefore it tends to pull Venus away from the earth. 

 When the moon is more distant from Venus than the earth is, the 

 attraction of Venus on the earth is the stronger, and tends to ])ull 

 it away from the moon, which, in regar'd to the disturbance of the 

 relative jilaces of the earth and moon, is the same thing as pulling 

 the moon away from the earth. In both these positions, therefore, 

 the disturbing force of Venus tends to pull the moon away from the 

 earth. When the earth and the moon are equally distant from Venus, 

 the attractions of Venus upon the two are equal, but not in parallel 

 lines ; the attractions tend to draw them along the sides of a wedge 

 whose point is at Venus, and, therefore, to diminish the distance 

 between them, or to push the moon towards the earth. 



Inasmuch as, in one pair of positions of the earth and moon, the 

 disturbing force of Venus tends to increase the distance between 

 them, and in another pair of positions it tends to diminish that di- 

 stance, it is important to ascertain which of these disturbances is the 

 greater. Suppose the distance of the moon from the earth to be 

 Y^ part of the distance of the earth from Venus. Then, when the 

 moon is between the earth and Venus, its distance from Venus is 

 yW of ^^^ whole ; the force upon it is '^^y* ^^ ^^^^ upon the earth ; 

 the excess of this (or the disturbing force tending to pull the moon 

 away from the earth) is -gV^' °^' nearly -^^ of that on the earth. In 

 like manner, when the moon is further from Venus than the earth 

 is, its distance from Venus is -Lg-^ of the earth's distance ; the force 

 upon it is ^g^g° of that upon the earth ; the defect of this (or the 

 disturbing force tending to pull the earth away from the moon) is 

 , gg' . or nearly -A- of that on the earth. But when the earth and 



10201' J 5 1 , -,. 



the moon are at equal distances from Venus, the proportion of then- 

 relative approach (as produced by the action of Venus) to the whole 

 effect of Venus upon them, is evidently represented by the inclina- 

 tion of the two lines drawn from them to Venus, or is the same as 

 the proportion of the distance of the moon from the earth, to the 

 distance of the earth from Venus, and is therefore yi^^ of the whole. 

 Thus the force tending to pull the moon from the earth at one time 

 is about double the force tending to push the moon towards the earth 

 at another time ; and therefore, upon the whole, the tendency of the 



