398 
MR. J. C. ADAMS ON THE SECULAR VARIATION 
3. Laplace’s explanation may be briefly stated as follows. He shows that the mean 
central disturbing force of the sun, by which the moon’s gravity towards the earth is 
diminished, depends not only on the sun’s mean distance, but also on the eccentricity 
of the earth’s orbit. Now this eccentricity is at present, and for many ages has been, 
diminishing, while the mean distance remains unaltered. In consequence of this 
the mean disturbing force is also diminishing, and therefore the moon’s gravity 
towards the earth at a given distance is, on the whole, increasing. Also, the area 
described in a given time by the moon about the earth is not affected by this alteration 
of the central force ; whence it readily follows that the moon’s mean distance from the 
earth will be diminished in the same ratio as the force at a given distance is increased, 
and that the mean angular motion will be increased in double the same ratio. 
4. This is the main principle of Laplace’s analytical method, in which he is followed 
by Damoiseau and Plana ; but it will be observed, that this reasoning supposes that 
the area described by the moon in a given time is not permanently altered, or in 
other words, that the tangential disturbing force produces no permanent effect. On 
examination, however, it will be found that this is not strictly true, and I will endeavour 
briefly to point out the manner in which the inequalities of the moon’s motion are 
modified by a gradual change of the central disturbing force, so as to give rise to 
such an alteration of the areal velocity. 
As an example, 1 will take the Variation, the most direct effect of the disturbing 
force. 
In the ordinary theory, the orbit of the moon as affected by this inequality only, 
would be symmetrical with respect to the line of conjunction with the sun, and the 
areal velocity generated while the moon was moving from quadrature to syzygy, 
would be exactly destroyed while it was moving from syzygy to quadrature, so that 
no permanent alteration of areal velocity would be produced. 
In reality, however, the magnitude of the disturbing force by which this inequality 
is caused, depends in some degree on the eccentricity of the earth’s orbit, and as this 
is continually diminishing, the central disturbing forces at equal angular distances 
on opposite sides of conjunction will not be exactly equal. Hence the orbit will no 
longer be symmetrically situated with respect to the line of conjunction. Now the 
change of areal velocityproduced by the tangential force at any point, depends partly on 
the value of the radius vector at that point, and consequently the effects of the tangential 
force before and after conjunction will no longer exactly balance each other. 
The other inequalities of the moon’s motion will be similarly modified, especially 
those which depend, more directly, on the eccentricity of the earth’s orbit, so that 
each of them gives rise to an uncompensated change of the areal velocity. 
Since the distortion in the form of the orbit just pointed out is due to the alter- 
ation of the disturbing force consequent upon a change in the eccentricity of the 
earth’s orbit, and it is by virtue of this distortion that the tangential force produces 
a permanent change in the rate of description of areas, it follows that this alteration 
