68 
PROFESSOR AIRY ON AN INEQUALITY OF LONG PERIOD 
implies the existence of an inequality whose period is about 240 years. No 
term has yet been calculated whose period is so long with respect to the 
periodic time of the planets disturbed *. The probability that there would be 
found some sensible irregularity depending on this term, may be estimated 
from this consideration; that in integrating the differential equations, this 
term receives a multiplier of 3 X 13 X (240) 2 , or about 2,200,000. 
On the other hand, the coefficient of this term is of the fifth order (with re¬ 
gard to the excentricities and inclinations of the orbits). The excentricities of 
both orbits are small. And it is remarkable that in the present position of the 
perihelia, the terms which would otherwise produce a large inequality destroy 
each other almost exactly. The inclination however is not so small; and upon 
this the existing inequality depends principally for its magnitude. 
The value of the principal term, calculated from the theory, I gave in a post¬ 
script to the paper above cited. I propose in the present memoir to give an 
account of the method of calculation, and to include other terms which are 
necessarily connected with the principal inequality. 
PART I. 
PERTURBATION OF THE EARTH’S LONGITUDE AND RADIUS VECTOR. 
Section 1. 
Method adopted for this investigation. 
1. The motion of a disturbed planet may be represented by supposing it to 
move, according to the laws of undisturbed motion, in an ellipse whose dimen¬ 
sions and position are continually changing: the epoch of the planet’s mean 
longitude at the origin of the time being also supposed to change. Putting a 
for the semi-axis major ; e for the excentricity; rs for the longitude of perihe¬ 
lion ; n for the mean motion in longitude in a unit of time ; s for the epoch, or 
the mean longitude when t — 0 ; (all which are variable) : m for the mass of 
the planet (Venus) ; g for the sum of the masses of the sun and planet; and the 
same letters with accents for the same quantities relative to another planet (the 
* The period of the long inequality of Saturn is only about thirty times as great as the periodic time 
of Saturn. 
