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bring his tables into accord with observation. In deter mi ning 
the correction to the secular acceleration employed by Hansen, 
it would appear that Professor Newcomb has attached consider- 
able weight to the very earliest set of observations in his list. 
It is almost certain, however, that there is some serious imper- 
fection in this set of observations, for they are entirely discordant 
with the remainder of the early observations, which are tolerably 
consistent with each other. As it is, Professor Newcomb has 
been unable to make the corrected tables represent this set of 
observations within nearly 16', whilst it very materially inter- 
feres with the other observations. For this reason this earliest 
set of observations may have proportionately less weight attached 
to it. 
Under these conditions it will be found that without ma- 
terially interfering with the accuracy with which the modem 
observations (1625-1875) are represented, the ancient obser- 
vations, including both the Arabian and the observations quoted 
by Ptolemy, can be much better represented by still further 
decreasing the value of the secular acceleration employed by 
Hansen. In fact, the value which best accords with the whole 
series of observations is 7*2", or within 1" of the theoretical 
value of the secular inequality. This reduces the supposed 
retardation of the rotation of the earth to a very slender foun- 
dation. 
Professor Newcomb points out the very important fact that 
the more recent observations indicate a value for the secular 
acceleration much less than that indicated by the earlier ones. 
In fact, the Arabian observations are best represented by the 
theoretical value of the secular acceleration, which is 6*2", while 
the ancient observations quoted by Ptolemy require the value 
7*4". This would seem to point to the existence of some 
periodical inequality. For some time it has appeared to me 
that the observations indicated the existence of a large term of 
very long period in the mean motion of the moon. This view 
is now confirmed by these investigations of Professor Newcomb. 
Such a term would require to have a period of over two thou- 
sand years to have a sensible effect on the secular acceleration. 
I have already shown that terms of this long period do exist in 
the mean motion of the moon. It remains to show that one of 
these terms has a coefficient of sufficient magnitude. Assuming 
the existence of such a term, and it can be shown that it would 
not only account for the difference between the values of the 
secular acceleration indicated by the modern and ancient obser- 
vations, but it would remove the remaining difference between 
the observed and theoretical values of the secular acceleration. 
For this purpose the term would have to have a period of about 
3,600 years, and a coefficient of under two minutes of arc. 
