THE GENERAL PERTURBATIONS OF THE MINOR PLANETS. 11 
CHAPTER I. 
Development of the Reciprocal of the Distance Between the Planets and its Odd 
Powers in Periodic Series. 
The action of one body on another under the influence of the law of gravitation 
is measured by the mass divided by the square of the distance. If then A be the dis- 
tance between any two bodies, this distance varying from one instant to another, it 
3 : : Wes : 
will be necessary to find a convenient expression for (5) in terms of the time. If 
r and 7’ be the radii-vectores of the two bodies, the accented letter always referring 
to the disturbing body, we have | 
N= 4+ r? — 2rr A. 
If we introduce the semi-major axes a, a’, which are constants, and their relation 
a’ S 
a=, we obtain 
= (3) + () @—2 (2) (2) (1) 
HT being the cosine of the angle formed by the radii-vectores. 
Let the origin of angles be taken at. the ascending node of the plane of the dis- 
turbed, on the plane of the disturbing, body. Let I, Il’, be the longitudes of the peri- 
helia measured from this point; also let f, 7’, be the true anomalies. The angle 
formed by the radii-vectores is (f’ + I’) —(f + Il); and the angles f + 0, f+ WW, 
being in different planes, we have 
H = cos (f + II) cos (f’ + Tl’) + cos Zsin (f ae iN Fsimy (G7 > UI), (2) 
I being the mutual inclination of the two planes. 
To find the values of I, Il’, J, let ® be the angular distance from the ascending 
node of the plane of the disturbed body on the fundamental plane to its ascending 
