186 
PROFESSOR G. B. AIRY ON THE ECLIPSES 
of some small inequalities and some small changes of coefficients, entirely insignificant 
in the computation of an ancient eclipse, they are the same as Damoiseau’s tables. 
20. The arguments which Damoiseau distinguishes by the letters u, x, t, 2 , 3 /, are 
formed in the following manner. To Damoiseau’s epochs, for the year in the nine- 
teenth century which differs by a whole number of centuries from the year for which 
the calculation is to be made, are added the numbers in his Table II. for the whole 
number of centuries taken backwards, and the corrections for u and 2 given at the 
foot of that table, and the corrections proportional to the square of the time in 
Table III., and the motions for 121^ 9 - 21-5 (to reduce Paris civil time to Greenwich 
astronomical time), and the numbers for the month, day, and hour of Greenwich 
mean solar time. The numbers thus formed are called Damoiseau’s Elements. It 
will be seen from this statement that I have adopted Damoiseau’s coefficients of the 
terms depending on the square of the time. Then the following corrections are added. 
For u (the mean longitude), the secular motion is increased by +1' 2T'-4, reckoning 
the years from 1814 ; the same correction is applied to t (the moon s elongation fioni 
the sun). For ^ (the mean anomaly), the secular motion is increased by +26"-4, 
reckoned from 1788. For?/ (the argument of latitude), the secular motion is increased 
by - 72 ^^- 8 , reckoned from 1782. These are called, in the subsequent articles, the 
Greenwich’ Corrections. By the addition of these, values of u, x, t, 2 , y are formed 
which are called the Unvaried Greenwich Elements, and these are the fundamental 
arguments used for the calculation of the moon’s places. The longitude and ecliptic 
polar distance thus found for a certain hour are altered by the horary motions found 
from the tables, and longitude and ecliptic polar distance are thus obtained for a second 
hour. These are converted into right ascension and north polar distance with the 
obliquity found in the solar calculations. The following error has been remarked m 
Damoiseau’s tables : Table IL, -2200 years, /or 113^*6634 read 113^-8634. 
21. With these right ascensions and declinations of the sun and moon, the circum- 
stances of the eclipse have been computed by the use of Woolhouse s methods in the 
Appendix to the Nautical Almanac, 1836. 
22. The next step is, to examine the effect of a possible change of elements. And 
here it may be remarked, that when the track of an eclipse is not highly inclined to 
the parallel upon the earth (which is true with regard to the eclipses here under con- 
sideration), a small change in the moon’s longitude produces little effect in the track 
of the eclipse. Partly for this reason, and partly because the place of node appears 
liable to the greatest uncertainty, I have recognized no error in the moon’s place 
except as depending on a possible error in the argument of latitude. In order to 
take account of this, I have in each calculation increased the argument of latitude by 
20' centesimal; and this changed element, taken in combination with the other 
elements unchanged, constitute the system which I call “Elements with Variation. 
With this new system of elements, the moon’s place is computed and the track of die 
central part of the shadow is computed, exactly as with the “ Unvaried Greenwich 
Elements.” The breadth of the shadow, as laid down on an ordinary chart, is 
