228 
MR. IVORY ON THE PERTURBATIONS OF THE PLANETS. 
a different method in equation (3) of the second section, and from which the 
value of h 2 , the semi-parameter of the variable elliptic orbit, was deduced. 
That element is therefore as much an immediate deduction from the disturbing 
forces, as is the mean distance in the equation of Lagrange. As the variation 
of a is the effect of the disturbing force in altering the velocity in the orbit, so 
the variation of h 2 is the effect of that part of the disturbing force which alters 
the exact proportionality to the times of the areas described round the sun. 
The two elements are together sufficient for determining both the form and 
the magnitude of the momentary elliptic orbit. The placing of this ellipse so 
as to be in intimate contact with the real path of the planet, a procedure 
which corresponds to finding the relation between the arcs 6 and v , determines 
the motion of the line of the apsides. 
If, lastly, we attend to that part of the disturbing force which is perpen- 
dicular to the circle of latitude passing through the planet, and proceed as 
before, we shall obtain the differential of the equation (4) in the second section. 
This differential is therefore the effect of the disturbing force in altering the 
momentary area which is described in the immoveable plane of x y, and which, 
without the action of this force would be proportional to the time. The elemen- 
tary area in the immoveable plane is the projection of the area described in the 
same time in the plane of the orbit ; the proportion of the two determines the 
cosine of the inclination of the variable plane in which the planet moves ; and 
from this it is easy to determine the position of the line of the nodes, as has 
been fully explained. 
What has been said is independent of the nature of the forces in action ; and 
it is obvious that the same method may be applied to estimate the effect of any 
extraneous force in disturbing the elliptic motion of a planet. 
It would appear that in the view we have taken of this problem, we have 
been making an approach to some general hints contained in the corollaries of 
the seventeenth proposition of the first book of the Principia. A connexion 
between the most recondite results of modern analytical science, and the 
original ideas thrown out by an author who, although he accomplished so 
much, has unavoidably left much to be supplied by his successors, is un- 
doubtedly worthy of being remarked, and may suggest useful reflections. 
December 22, 1831. 
