TRANSACTIONS OF SECTION A. 313 
Cette question est donc de la derniére importance et il sera toujours bon d’en 
approfondir toutes les difficultés, avant qu’on en puisse espérer une solution 
complete.’ x 
In the final results of my work the development aims to include the gravita- 
tional action of every particle of matter which can have a sensible effect on the 
moon’s motion, so that any differences which appear between theory and observa- 
tion may not be set down to want of accuracy in the completeness with which 
the theory is carried out. Every known force capable of calculation is included. 
So much for the theory. Gravitation, however, is only a law of force: we 
need the initial position, speed, and direction of motion. To get this with 
sufficient accuracy no single set of observations will serve; the new theory must 
be compared with as great a number of these as possible. To do this directly 
from the theory is far too long a task, and, moreover, it is not necessary. In the 
past every observation has been compared with the place shown in the ‘ Nautical 
Almanac’ and the small differences between them have been recorded from day 
to day. By taking many of these differences and reducing them so as fo 
correspond with differences at one date, the position of the moon at that date 
can be found with far greater accuracy than could be obtained through any one 
observation. At the Greenwich Observatory the moon has been observed and 
recorded regularly since 1750. With some 120 observations a year, there are 
about 20,000 available for comparison, quite apart from shorter series at other 
observatories. Unfortunately these observations are compared with incorrect 
theories, and, in the early days, the observers were not able to find out, with the 
accuracy required to-day, the errors of their instruments or the places of the 
stars with which the moon was compared. But we have means of correcting the 
observations, so that they can be freed from many of the errors present in the 
results which were published at the time the observations were made. We can 
also correct the older theories. They can be compared with the new theory and 
the differences calculated: these differences need not even be applied to the 
separate observations, but only to the observations combined into properly chosen 
groups. Thus the labour involved in making use of the earlier observations is 
much less than might appear at first sight. 
For the past eighteen months I have been engaged in this work of finding the 
differences between the old theories and my own, as well as in correcting those 
observations which were made at times before the resources of the astronomer 
had reached their present stage of perfection. I have not dealt with the observa- 
tions from the start : other workers, notably Airy in the last century and Cowell 
in this, have done the greater part of the labour. My share was mainly to 
carry theirs a stage further by adopting the latest theory and the best modern 
practice for the reduction of the observations. In this way a much closer agree- 
ment between theory and observation has been obtained, and the initial position 
and velocity of the moon at a given date are now known with an accuracy com- 
parable with that of the theory. I shall shortly return to this problem and 
exhibit this degree of accuracy by means of some diagrams which will be thrown 
on the screen. 
I have spoken of the determination of these initial values as if it con- 
stituted a problem separate from the theory. Theoretically it is so, but prac- 
tically the two must go together. The increase in accuracy. of the theory 
has gone on successively with increase in accuracy of the determination of 
these constants. We do not find, with a new theory, the new constants from 
the start, but corrections to the previously adopted values of these constants. 
In fact, all the problems of which I am talking are so much inter-related that 
it is only justifiable to separate them for the purposes of exposition. 
Let us suppose that the theory and these constants have been found in 
numerical form, so that the position of the moon is shown by means of 
expressions which contain nothing unknown but the time. To find the moon’s 
place at any date we have then only to insert that date and to perform the 
necessary numerical calculations. This is not done directly, on account of the 
labour involved. What are known as ‘Tables of the Moon’s Motion’ are 
formed. These tables constitute an intermediate step between the theory and 
the positions of the moon which are printed in the ‘Nautical Almanac.’ 
Their sole use and necessity is the abbreviation of the work of calculation 
