THE MOON’S FACE. 
267 
the case of moonlets. As the moonlets, by postulate, moved 
initially in orbits not very dissimilar from that of the moon 
and in the same direction, their initial velocities with refer¬ 
ence to the moon were small as compared with the velocities 
created by the moon’s attraction. Their courses in the 
near vicinity of the moon were therefore essentially parts of 
curved orbits of the nature of conic sections, with the moon 
at the focus, and could not properly be treated as equably 
distributed straight lines. Furthermore, the initial veloci¬ 
ties of moonlets with reference to the moon, that is, the ve¬ 
locities with which they overtook or were overtaken by the 
moon, were not all the same, but were varied in a systematic 
way, being greater in proportion as the orbits of the moon¬ 
lets differed from the moon’s orbit.* Account was taken of 
these relations, but the influence of the earth’s attraction, 
essential to a rigorous discussion, was ignored ; the orbital 
equation of the moonlets was derived, the conditions of col¬ 
lision with the moon were examined, and the general ex¬ 
pression for the angle of incidence was obtained.f This 
general expression is— 
n — V sin i, 
* Assuming circular orbits for moon and moonlet, calling their distances 
from the earth D and d, and calling their velocities in their orbits V and 
V + u, we have, from Kepler’s third law, 
J (V+u) Dtt\* _ (dV 
X VdTT / —\d) 
whence 
D--d_ 2F+« 
— d 
Since, in the case of any colliding moonlet, u is very small in compari¬ 
son with 2 V, the fraction in the second member is sensibly constant, 
and u, the relative velocity of moonlet and moon, varies as —or 
approximately as D — d, the distance between the orbits. 
fAt a distance c from the moon’s center a moonlet has the initial 
velocity u. Placing the origin at the moon’s center and the axis of ref¬ 
erence parallel to the direction of the initial velocity, the polar equation 
to the moonlet’s orbit is— 
_ b 2 u 2 /v 
1+ <L cose -(b-M) sine’ ■ 
c \ c n } 
where p is the radius vector, 6 its inclination to the axis of reference, b the 
length of a perpendicular drawn to. the axis from the point where the 
