S. Newcomb—Mean Motion of the Moon. 407 
In column (1) we have the mean correction indicated by 
observations to Hansen’s tables of the moon without any mod- 
ification whatever. In column (2) these corrections are modi- 
fied by the effect of Hansen’s empirical term, so as to show the 
corrections to the pure theory after this term is subtracted 
from the tables. If "the theory is perfect, these numbers ought 
The —— are the several corrections given by the method 
of least square 
dex 41957 
dn= —12 id: URpoek, 1700, 
ds= — 3° 
The value of the secular eae. adopted by Hansen is 
12°17. Subtracting the correction it seems that the accelera- 
tion to which we are led by observation alone, is 
Column (3) shows the outstanding corrections whiol remain 
value of the see acceleration much less than that indi- 
cated by the older ones. If we investigate the uniform varia- 
tion of the siesibebition which would best satisfy the whole of 
the observations, we shall find it to be —0”°9 in a century. 
The hypothesis of such a uniform vwiedon is, however, too 
improbable to be admitted; and moreover, it still fails to 
represent the modern observations, although the ancient ones 
are thus greatly improve 
In recent times it has been generally considered that the 
difference between the theoretical acceleration and that given 
by observations arises from a change in the length of the 
ay. It is worthy of remark that by supposing this change 
itself subject to variations, all the apparent changes in the 
mean motion of the moon can be accounted for. This is a 
hypothesis which I have suggested in former numbers of this 
the 
between observation and aioe In the first place, the sec- 
ular acceleration must be supposed to be "iilonn and equal to 
617. Two epochs at which we may suppose the time given 
by the rotation of the earth to be correct, being entirely 
arbitrary, we shall take 1750 pe ge 0 for these epochs. 
Having thus formed a theory of the moon’s mean motion 
ary. 
founded on gravitation alone, column (4). cone the apparent 
corrections indicated by observation. In column (5) these cor- 
rections are changed into time. The times here given are 
