450 MR. G. H. DARWIN ON THE PRECESSION OF A VISCOUS SPHEROID, 
Let m be the moon’s mass, c her distance, and 7 =#-. 
1 $ 
Let XYZ (Plate 36, fig. 1) be rectangular axes fixed in space, XY being the ecliptic. 
Let M be the moon in her orbit moving from Y towards X, with an angular 
velocity ft. 
Let ABC be rectangular axes fixed in the earth, AB being the equator. 
Let ?, xfi be the coordinates of the pole C referred to XYZ, so that i is the obliquity 
of the ecliptic, and the precession of the equinoxes. 
Let 1 , 0, (f) be the polar coordinates of any point P in the earth referred to ABC, 
as indicated in the figure. 
© 
Let ctq, Wo, oq be the component angular velocities of the earth about the instan¬ 
taneous positions of ABC. 
Then we have, as usual, the geometrical equations, 
di 
dt 
— — oq sin y + on cos y 
d/Jr . 
Yt 8111 1 = 
cos ;— 
dt + dt C08 
■co 
O), 
l cos y — Wo sin y 
. . . ( 1 ) 
J 
Let n cosec i be the precession of the equinoxes, or , so that = II cot i—a> s * 
Now the earth rotates with a negative angular velocity, that is from B to A; therefore 
cJ/'V • 
if we put -jy=n, n is equal to the true angular velocity of the earth +11 cot i. But for 
CLL 
purposes of numerical calculation n may be taken as the earth’s angular velocity; and 
care need merely be taken that inequalities of very long period are not mistaken for 
secular changes. 
Let the epoch be taken as the time when the colure ZC was in the plane of ZX, 
when y was zero and the moon on the equator at Y. It will be convenient also to assume 
later that there was also an eclipse at the same instant. A number of troublesome 
symbols are thus got rid of, whilst the generality of the solution is unaffected. 
7T 77" 
Then by the previous definitions we have y —nt, MN=/2£, NB = q — BD=- — (<£—y). 
Now if iv be the mass of the homogeneous earth per unit volume, then the tide¬ 
generating gravitation potential V of the moon, estimated per unit volume, at the 
point r, 6, <j> or P in the earth is, by the well-known formula, V=t0rr 3 (cos 3 PM—-§■). 
This is the function on which the tides depend, and as above explained, it must be 
* The limit of Tl cot i is still small when i is zero. In considering the precession with one disturbing 
body only, n cosec L is merely the precession due to that body; but afterwards when the effect of the sun 
is added it must be taken as the full precession. 
