OF THE PERTURBATIONS OF THE PLANETS. 
217 
the expressions of dx, dy, dz will contain d £ only: for dv contains d £ only, 
and is independent of the differentials of s, e, sr. Thus we have 
j d x j fy dx. 
Til 
d % 
— dt dt* 
dt 
It is in these properties that we recognise the principle of the V xriation of 
the arbitrary constants. The finite expressions of x, y , z, being the same in the 
immoveable ellipse described by the sole action of the centripetal force of the 
sun, and in the variable ellipse which represents the motion of a disturbed 
planet, they will verify the equations (A), supposing the arbitrary quantities 
constant, and neglecting the disturbing forces. The velocities are 
the same whether the arbitrary quantities remain constant or vary ; and thus, 
for a moment of time d t, the motion in the invariable ellipse coincides with 
that of the planet in its real path. But, in the next moment of time, the planet 
will quit the periphery of the ellipse supposed to continue invariable ; because 
the forces in that orbit are different from the forces which urge the planet. In 
the immoveable ellipse the forces in the directions of the coordinates are equal 
to ar bitrary quantities being constant ; but, in the case of 
the planet, the like forces are equal to the same differentials augmented by 
the variation of the arbitrary quantities, the additions thus introduced being 
equal to the disturbing forces, (k (k [k It is in this manner that an 
elliptic orbit, by the variation of its elements, is capable of representing at 
every moment of time both the velocity of a disturbed planet, and the forces 
by which it is urged. 
And generally, when a dynamical problem admits of an exact solution, the 
arbitrary quantities may be made to vary so as not to alter the velocities 
d v dy d % 
jj, f t , ^7 ; an d the additions which the variation of the same quantities makes 
. . ddx ddy ddz .... 
to the expressions will represent new forces introduced in the 
problem. By means of this artifice we may estimate the effect of any disturb- 
ing forces, more especially of such as bear an inconsiderable proportion to the 
principal forces, in altering the original motion of the body. This is the prin- 
mdcccxxxii. 2 F 
