76 A NEW METHOD OF DETERMINING 
In a similar manner we find 
dy cos f —d@asin f =r df + 2dr df. 
Operating on equations (2) in the same way, we have 
S 3 COS f + 5 sin f + Eee = X.cosf+ Y.snf=F 
a cos f — sin f = Y.cosof—X sn f=S8S 
Comparing the two sets of equations, we have 
ai 
ref, odr af 3 1 
ap 0 Sap ar =h (1+ m), 
(3) 
ir ay” ed+tm) — 7. 
a aa 
The second members of equations (1) and (2) are small, and in a first approxi- 
mation to the motion of m relative to the Sun, we can neglect them. The integration 
of equations (2) introduces six arbitrary constants; and the integration of equations 
(3) introduces four. These constants are the elements which determine the undis- 
turbed motion of m around the Sun. Having these elements, let 
a) the semi-major axis, 
nm the mean motion, 
go) the mean anomaly for the instant ¢ = 0, 
é the eccentricity, 
$) the angle of eccentricity, — 
nm, the angle between the axis of « and the perihelion, 
v, the angle between the axis of x and the radius-vector, 
jo the true anomaly, 
é the eccentric anomaly. 
These elements are constants, and give the position of the body for the epoch, or 
fort=0. Let us now take a system of variable elements, functions of the time, and 
let them be designated as before, omitting the subscript zero, and writing x in place 
