OF AGATHOCLES, THALES, AND XERXES. 
m 
from that which would have been found if they had not been considered. This 
change is fully recognized in the paper in volume xvii. of the Royal Astronomical 
Society’s Memoirs. 
12. Secondly, as to the correction to the motion of the moon’s node. The com- 
parison, between an observed latitude of the moon (supposed near the node) and the 
corresponding tabular latitude of the moon as calculated from Damoiseau’s tables, 
gives in fact a comparison between the true argument of latitude and Damoiseau’s 
argument of latitude ; and therefore it gives the sum of two corrections to two 
different elements of Damoiseau’s tables; namely, the correction to Damoiseau’s 
longitude of the moon, and the correction to Damoiseau’s supplement of longitude of 
moon’s node. And when, from the groups of results, we infer the secular correction 
to the motion of the argument of latitude^ we have found the sum of two corrections ; 
first, the secular correction to Damoiseau’s mean motion of the moon, such as will 
best reconcile Damoiseau’s longitude of the moon, unaffected by new inequalities, 
with the observed longitude of the moon, during the period to which the reductions 
apply; and, secondly, the secular correction to Damoiseau’s regression of the node. 
13. Now for the difference between Damoiseau’s longitude unaffected by new 
inequalities, and the observed longitude, we must refer to the second column of the 
table. Royal Astronomical Society’s Memoirs, vol. xvii. p. 35 ; and, comparing the 
first four numbers with the last four, we find the correction to Damoiseau’s 
motion of the moon during 38T years = — 0"’30. But in page 64, the correction to 
Damoiseau’s motion of argument of latitude during the same time is — 24"‘26. 
Hence the correction to Damoiseau’s regression of the node during that time is 
— 24"'26 + 0' '30= — 23"’96, or the secular correction is — 62"'9. Applying this to 
Damoiseau’s secular regression in a Julian century (namely, 134° 9' 57"‘5), we find 
the corrected regression 134° 8' 54"'6, which is probably the most accurate that can 
yet be deduced from meridional observations. 
14. But, in correcting Damoiseau’s tables for further use, we must not apply the 
quantity — 62"'9 to his secular motion of argument of latitude. The moon’s tabular 
longitude at any time, putting H for the value of Hansen’s inequalities at that time, 
will require (see page 36 of the same Memoir) the correction -f-39"'3x number of 
centuries H-H ; and as the tabular supplement of node requires the correction — 62"'9 X 
number of centuries, the tabular argument of latitude (which is the sum of moon’s 
longitude and supplement of node) will require a correction equal to the sum of these 
two corrections, or — 23"'6x number of centuries +H. For a distant eclipse, the 
quantity H may be safely neglected. 
15. Thirdly, as to the correction to the motion of the moon’s perigee. The same 
considerations, in every respect, which have been used for determining the correction 
to the motion of node and to the motion of argument of latitude, are to be used for 
determining the correction to the motion of perigee and to the motion of mean 
anomaly. Thus, in p. 40 of the Memoir above cited, the apparent correction of motion 
