L = nt + L0.
Such an angle continually goes through the round of 360° in a definite period. For example, if the daily motion is 5°, and we take the day as the unit of time, the round will be completed in 72 days, and the angle will continually go through the value which it had 72 days before. Let us now consider an equation of the form
U = a sin (nt + L0).
The value of U will continually oscillate between the extreme values +a and −a, going through a series of changes in the same period in which the angle nt + L0 goes through a revolution. In this case the variation will be simply periodic.
The value of any element of the planet’s motion will generally be represented by the sum of an infinite series of such periodic quantities, having different periods. For example
U = a sin (nt + L0) + b sin (mt + L1) + c sin (kt + L2) &c.
In this case the motion of U, while still periodic, is seemingly irregular, being much like that of a pitching ship, which has no one unvarying period.
In the problems of celestial mechanics the angles within the parentheses are represented by sums or differences of multiples of the mean longitudes of the planets as they move round their orbits. If l be the mean longitude of the planet whose motion we are considering, and l′ that of the attracting planet affecting it, the periodic inequalities of the elements as well as of the co-ordinates of the attracted planet, may be represented by an infinite series of terms like the following:—
a sin (l′ − l) + b sin (2l′ − l) + c sin (l′ − 2l) + &c.
Here the coefficients of l and l′ may separately take all integral values, though as a general rule the coefficients a, b, c, &c. diminish rapidly when these coefficients become large, so that only small values have to be considered.