CONTRIBUTIONS TO THE THEORY OF THE MOTION OF THE MOON. 237 
a third calculation. It is understood that this third investiga- 
tion was never completed. It is well known, however, that 
Hansen would not admit the correctness of Delaunay’s results, 
although he never stated his reasons for considering them to he 
imperfect. Little importance, however, seems to have been 
attached to Hansen’s refusal to admit that Delaunay’s researches 
invalidated his own, because it seemed natural to suppose that 
had Delaunay’s investigations been imperfect, Hansen would 
have at once pointed this out. It has always seemed to me that 
Hansen must have had some very good reason for his objection 
to Delaunay’s results, but that he was waiting until he had 
completed his third investigation of this difficult problem. 
Moreover it has already been mentioned that Delaunay employed 
his own -peculiar method of attacking the problem, so that 
one who was not perfectly familiar with this new method would 
not readily follow the research. The last contribution to this 
subject opens the question — are Delaunay’s investigations com- 
plete ? This is a most important matter, for if Delaunay has 
omitted to take into consideration the whole of the problem, it 
is easy to see not only why there should be a difference between 
Hansen’s and Delaunay’s results, but why Hansen refused to 
admit the accuracy of Delaunay. 
In his investigations Delaunay has very carefully calculated 
the effect of the attraction of Venus on the moon, and the 
effect of Venus upon the earth, and in this way indirect effect 
on the moon. The manner in which the planet thus produces 
inequalities in the motion of the moon has been already 
detailed. In these calculations Delaunay has carefully taken 
into consideration the fact that, owing to the difference in the 
attraction of the sun upon the moon and earth, the moon in its 
motion around the earth is affected by numerous inequalities. 
In all this work Delaunay’s results are perfectly accurate. Be- 
yond this Delaunay has not gone, and this really comprises the 
entire direct way in which Venus influences the motion of the 
moon. It has also been shown that from this action of the 
planet Venus for over a hundred years at a time, the moon 
moves in a very slightly larger orbit than it could were there no 
Venus to attract it, and that this is then followed by a similar 
period in which the moon moves in a proportionately smaller 
orbit. What Delaunay has not taken into consideration is the 
way in which the action of the sun upon the moon will be 
affected by this alternate enlargement and contraction of the 
orbit of the moon arising from the action of Venus. It is known 
tha/t the slower the moon revolves around the earth, the more 
powerful is the disturbing action of the sun and the greater is 
the power it has of enlarging the orbit of the moon. During 
the period, therefore, when the attraction of the planet Venus 
