OF THE PERTURBATIONS OF THE PLANETS. 
209 
and, by substituting- the value of + s, 
and, observing that r 2 dX= (1 s 2 ) li' dt = (1 + s 2 ) h cos idt sj 'p, we 
finally get, 
These equations determine the motion of the node in longitude, and the varia 
tion in the inclination of the orbit. They are rigorously exact, and may be 
transformed in various ways, as it may suit the purpose of the inquirer. 
We proceed now to investigate the motion of the planet round the sun. 
For this purpose we have the equations (2) and (3), viz. 
And first, as the small arc dv contained between the two radii r and r + dr, 
continually passes from one plane to another, it is requisite to inquire what 
notion we must affix to the sum v. The momentary plane of the planet’s 
motion, in shifting its place, turns upon a radius vector ; and if we suppose a 
circle concentric with the sun to be described in it, and to remain firmly at- 
tached to it, the differentials dv will evidently accumulate upon the circum- 
ference of the circle, and will form a continuous sum, in the same manner as 
if the plane remained motionless in one position. The arc v is therefore the 
angular motion of the planet round the sun in the moveable plane, and is 
reckoned upon the circumference of the circle from an arbitrary origin. 
In the first of the foregoing equations a is an arbitrary constant, and I shall 
put. 
r 2 dv = hdt Jp. 
so that we shall have 
dr a r % dv 2 1 
r ' a 5 
r 2 dv = hdt [>j ; 
2 E 
MDCCCXXXII. 
