234 NOTE ON THE INEQUALITY IN THE MOON'S LATITUDE [29 



Moon's orbit with it, so that the inclination of the Moon's orbit to the 

 variable ecliptic is not liable to any secular variation. 



In the same place he finds an analytical expression for the perturbation 

 of latitude in reference to the variable ecliptic which is caused by the 

 secular change in that plane. 



Now the point to be noticed is that this analytical expression given 

 by Laplace requires only the very slightest possible development to furnish 

 for the inequality in question a result which is identical with the value 

 given by the formula of Hansen, in which displacements of the ecliptic 

 varying not only as the first but also as the second power of the time 

 are taken into account. It is true that Laplace imagined that this in- 

 equality would turn out to be insensible, but this was only because he had 

 not attempted to turn his formula into numbers. 



Analysis. 



I. Investigation of the inequality in the Moon's latitude which is due 

 to the secular motion of the plane of the ecliptic, making the same sup- 

 positions and employing the same data as the Astronomer Royal. 



At the time t let x, y, z be the rectangular coordinates of the Moon, 

 and x', ]f those of the Sun, referred to the Earth's centre as origin, the 

 variable plane of the ecliptic at the same time being taken as the plane 

 of xy. 



Also at the time t let , 17, be the rectangular coordinates of the 

 Moon, and ', >/, ' those of the Sun, taking the fixed plane of the ecliptic 

 corresponding to t as the plane of r). 



For greater simplicity we will suppose, with the Astronomer Royal, 

 that the variable ecliptic intersects the fixed ecliptic in a fixed line, and 

 that the angle between these two planes is proportional to the time. 



Let this fixed line be taken as the axis of x and also as the axis 

 of f, and let cat be the angle between the variable and the fixed ecliptic, 

 then the relations between the coordinates belonging to the two systems 

 will be 



if = X, 



t]\l cos (at z sin ant, 

 z cos (at + y sin a>t, 



