212] A MEMOIR ON THE PROBLEM OF DISTURBED ELLIPTIC MOTION. 271 
thus a seventh variable element, the departure of the node. The element just referred 
to (the departure of the node) forms, with the longitude of the node and the inclination, 
a group of three elements, which determine the position of the orbit and of the 
departure-point. The coordinates of the planet are in the first instance taken to be 
the radius vector, longitude, and latitude ; but the before-mentioned three elements 
being considered as given, the position of the planet depends only on the radius vector 
and the departure. These may be then expressed in terms of the remaining four 
elements ; as to the choice of these four elements, it is to be remarked that there is 
one element which only enters through the mean anomaly, and that there is great con 
venience in representing with Hansen the mean anomaly by a single letter ; and that in 
the various formulae we may use, in the place of the element implicitly involved in the 
mean anomaly, the mean anomaly itself, or treat the mean anomaly as an element ; the 
four elements may be taken to be the semi-axis major, the eccentricity, the mean 
anomaly, and the departure of the pericentre. And joining to these the before-mentioned 
three elements, we have the system of elements represented in the memoir by 
a, e, g, -sr, a, 0, <£. It has been assumed so far that the three elements determine 
the position of the orbit and departure-point in reference to a fixed plane and origin of 
longitudes ; but we may suppose more generally that, instead of the fixed plane and 
origin of longitudes, we have a variable plane or orbit of reference and a departure- 
point in this variable orbit of reference. The quantities which determine the orbit of 
reference and departure-point are naturally taken to be the departure of the node, 
longitude of the node, and inclination ; these are assumed to be given functions of the 
time, and they are in the memoir represented by a', O', The three elements of the 
planet’s orbit (viz. departure of node, longitude of node, and inclination) in relation to 
the orbit of reference and departure-point therein, are in the memoir represented by 
X, ©, T>, and the system of elements ultimately adopted is therefore a, e, g, ot, X, ©, T>. 
I obtain formulae for the variations of these elements under two different modes of 
expression of the disturbing function : first, when the disturbing function is expressed 
in terms of the radius -vector and departure and of the three elements X, ©, : 
secondly, when the disturbing function is expressed in terms of the seven elements 
a, e, g, sr, X, ©, <É>. The establishment of the two sets of formulae just referred to 
constitutes the chief object of the memoir; but the memoir contains some other inves 
tigations and formulae in relation to the general subject. 
The coordinates of the planet are 
r, the radius vector, 
v, the longitude, 
y, the latitude. 
The attractive force at distance unity is for convenience represented by n 2 a\ which 
denotes, therefore, an absolute constant ; but the significations of n and a are not yet 
defined. 
The disturbing function, as used by Lagrange, is denoted by il, that is fl = — R, 
if R be the disturbing function of the Mécanique Celeste.