24 
MR. LUBBOCK’S RESEARCHES 
tion of rotation is commensurate to the mean motion of revolution of the 
luminary M 1 . 
^ = series of sines without any constant quantity, except in a similar case. 
c being equal to a constant + a series of sines. 
= a series of sines without any constant quantity. 
^ = a series of cosines + a constant quantity. 
d t 
- — a series of cosines + a constant quantity. 
d t 
In the general case where A is not equal to B, n = constant + series of 
cosines. 
The form of the preceding expressions is not affected however, for the ap- 
proximation may be carried so that except in the case of commensurability 
above mentioned, the mean motion of rotation being also nearly twice the 
mean motion of the planet 71/' in its orbit or greater, 
n = constant + series of cosines without any constant quantity multi- 
plied by the time. 
c = constant series of sines or cosines without any constant quantity 
multiplied by the time. 
a = constant -j- series of cosines without any constant quantity multi- 
plied by the time. 
y = constant + series of cosines or sines + a constant quantity multi- 
plied by the time. 
ipo = constant -f- series of sines + a constant quantity multiplied by the 
time. 
<p 0 = constant + series of sines + a constant quantity multiplied by the 
time. 
The constant quantity multiplied by the time in the value of is the pre- 
cession of the equinox. 
If z 1 = 0, (which amounts to taking for the fixed plane the orbit of the 
planet 71/',) and n 1 be its mean motion, then neglecting the eccentricity, 
x 1 = a cos n 1 1, y' = a! sin ri t, r 1 = 
n 3 | | a J}/ 3 • * o f 4. 
P = jj sin w cos uj (1 — cos 2 n t), P = — sin cu sin 2 n' t 
