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the earth, and at the end of the one hundred and twenty years 
has gained an appreciable amount, whilst for the one hundred 
and twenty years when the earth is nearer the sun the reverse 
phenomenon takes place. This term may be called the indirect 
term of long period. The importance of these terms arises from 
their effects being cumulative for so many years, that although 
very small, at the end of the long period in which they act in 
one way they accumulate to something considerable. 
It was generally considered by astronomers that Professor 
Hansen had succeeded by this means in removing the last 
difficulty in the theory of the moon, and Sir John Herschel 
remarked in his address to the Royal Astronomical Society, 
66 that their discovery may be considered as a practical comple- 
tion of the lunar theory, at least for the present astronomical 
age, and as establishing the entire dominion of the Newtonian 
theory and its analytical applications over that refractory satel- 
lite.” This view received full confirmation when it was found 
that Hansen’s “Tables de la Lune,” based on his investigation, 
accurately represented the motion of the moon. It appeared 
certain that the difficulties of the lunar theory were at last 
mastered. 
Unfortunately this sanguine view has been found to be pre- 
mature ; unexpected difficulties have sprung up in the lunar 
theory, and Hansen’s 66 Tables de la Lune,” which so well repre- 
sented the place of the moon during the interval 1750-1850, 
have become seriously inaccurate, and are rapidly becoming 
more and more erroneous. It is certain, therefore, that there is 
still some serious deficiency in the theory of the moon which 
formed the basis of Hansen’s “ Tables de la Lune.” It is there- 
fore a matter of the greatest importance that the lunar theory 
should be rendered more perfect and the tables of the moon 
made more accurate. The practical importance of this is so 
great that it is probable that before long steps will be taken by 
the authorities of both the American and English Nautical 
Almanacs to empirically correct the tables, so as to render the 
tabular places more accurate. 
It is now generally known that Hansen, to make his tables 
represent the motion of the moon during the interval 1750- 
1850, was obliged to give empirical values to the coefficients of 
the two terms of long period which he had discovered to be pro- 
duced by the action of Venus. Hansen considered he was 
justified in doing this by the circumstance that these empirical 
values enabled his tables to represent with accuracy the motion 
of the moon during the century prior to 1850, for it appeared 
most improbable that the tables could represent the motion of 
the moon for a whole century and yet fail to continue to repre- 
sent it. The tables, however, did not continue to represent the 
