CONTRIBUTIONS TO THE THEORY OF THE MOTION OF THE MOON. 233 
motion of the moon, and have been steadily deviating from the 
truth, until at the present time they show most serious and in- 
creasing errors. It is, therefore, certain that at least one, if not 
both of the empirical values which Hansen assigned to these two 
terms of long period must be erroneous. What then are the 
correct values, as Hansen’s are wrong ? Many of the recent 
contributions to the lunar theory have implicitly turned on this 
point. 
The late M. Delaunay was the first to take up this subject, 
for he was unsatisfied with the manner in which Professor 
Hansen had dealt with these terms of long period. Delaunay 
had then been engaged for ten years on the theory of the moon, 
and he had devised an entirely original method of attacking the 
problem. This new method which he employed was most 
elegant and ingenious. Instead of attacking the problem all at 
once, he showed that it was possible to divide it into several 
hundred simpler problems, and to solve each of these separately. 
Each of the many hundred different perturbations to which the 
motion of the moon is subject was taken by itself and its in- 
fluence separately calculated. By this means he had to deal 
with over a thousand simple operations, instead of two or three 
dozen much more complex operations. This new method is in 
practice very laborious, it is very liable to errors, and it requires 
much skill and experience in its use, but its analytical elegance 
and advantages won a high reputation for M. Delaunay. More- 
over, in his skilled hands it proved a most successful means of 
attacking the problem, for he was enabled by its means to carry 
the lunar theory to a degree of perfection far beyond any of his 
predecessors. 
Delaunay applied this method to the determination of the 
values of the two terms of long period which had been discovered 
by Hansen to be produced by the attraction of the planet 
Venus. He considered, first, the one which is due to the direct 
action of Venus on the moon ; and in his calculations he neg- 
lected the effect of the small inclination of the orbit of Venus, 
because he considered that it would not be of any consequence. 
He found that this first term of long period would be very small, 
and he announced that Hansen’s value was more than fifty times 
too large. He was induced, however, to re-examine his inves- 
tigations, and, to render them somewhat more complete, he took 
into account the small inclination of the orbit of Venus. This 
new result he found very different from the previous, for it 
agreed closely with the value found by Hansen. The small 
inclination of the orbit of Venus which he had thought to be of 
no importance proved to be the very cause why the attraction 
of Venus was able to so much disturb the motion of the moon. 
The value for this term of long period which Delaunay had thus 
