184 
PROFESSOR G. B. AIRY ON THE ECLIPSES 
of anomaly in 38-1 years is -12"-03: by the second column of the table on p. 35, 
the apparent correction of mean motion in the same time (the tabular values being 
unaffected by Hansen’s inequalities) is -0"-30 ; hence the true correction to the mo- 
tion of perigee (estimated as a regression) in 38-1 years is — 12"*03+0"-30=~ll"-/3. 
And the true correction to the moon’s motion of mean longitude in 38‘1 \ ears is 
4-14"-99. Therefore the true correction to the motion of the mean anomaly in 38-1 
years is -}-l 4 "* 99 — ll"73=-l-3"-26 ; and the true secular correction is -f-8"-56. This 
supposes that Hansen’s inequalities in longitude are not accompanied with sensible 
inequalities in the place of perigee. 
16. The position of the moon at any time, as affecting the circumstances of an eclipse, 
will depend on the moon’s mean longitude, the longitude of perigee, and the longitude 
of node. The values of these three elements for any late year are known with very 
great accuracy (their values for certain years are given in the Memoir repeatedly 
cited) ; and the annual motions of mean longitude and longitude of perigee for a 
Julian century at the present time are very accurately known ; in that of the longi- 
tude of node there is a very minute uncertainty. But the secular motion of each of 
these elements changes from century to century ; and terms are thus introduced into 
the expression for each of these elements depending on the square and higher powers 
of the time. Laplace was the first who computed (from theory) the coefficients of 
these terms ; and his numbers were adopted, with insignificant alterations, in Burg s 
and Burckhardt’s tables. Damoiseau, on repeating the investigation, obtained 
different values for the coefficients ; in particular, he introduced in the. coefficient 
which relates to the place of perigee a change of such magnitude as very greatly to 
modify the circumstances of any calculated distant eclipse. Plana and Hansen, b\ 
independent investigations conducted in different ways, have in general confiimed 
these alterations ; the result, however, of Hansen’s last investigation- differs some- 
what from that of his former investigation, though by a very much smaller quantity 
than the difference of each from Laplace’s values. I shall give here the coefficients 
of the square of the number of centuries obtained by these writers ; the signs of those 
numbers which relate to perigee and node being applicable to progression of 
perigee and regression of node. The reader must remark that a change of 1 in the 
coefficient for mean longitude, of 9" in that for longitude of perigee, or of 11" in that 
for longitude of node, produces, in the moon’s place for a perigeal eclipse at the time 
of Thales, an effect of about 10', and that this will alter the place of the eclipse- 
shadow at a given time not less than 10° on the earth’s surface. 
Laplace, Mecanique CHeste, vol. iii. pages 237, 273, 274. 
Coefficient for mean longitude -]-10T8 
Coefficient for longitude of perigee — 30'55 
Coefficient for suppl. longitude of node — 7*49 
