268 
GILBERT. 
n being the relative number of colliding moonlets whose 
angle of incidence is less than i. It indicates that 58 per 
cent, deviate less than 20° from the vertical, 70 per cent, 
less than 30°, and 80 per cent, less than 40°; and it yields 
the distribution curve marked C in Fig. 12.* 
The theoretic distribution obtained by this partial treat¬ 
ment accords so well with the phenomena under discussion 
velocity is u, a the distance from the foot of that perpendicular to the 
origin, and n the constant of gravitation for the moon. 
The radius of the moon being r, the condition that the moonlet collides 
with the moon is— 
The general expression for the angle of incidence, i, is— 
sm i = 
b 
r \ 
M) 
in which 2 n/r == V 2 — the square of the velocity acquired by a body 
falling to the moon from an infinite distance. Since r/c is a small frac¬ 
tion and 2 fij ru 2 is a large number— 
sin i = (nearly). 
rV 
By postulate (preceding note) u varies as b, and since r and V are both 
constant— 
b — j/ sin i X constant. 
* The curves A, B, and C of Fig. 12 represent the distribution of inci¬ 
dent bodies with reference to angles of incidence under the laws expressed 
severally in the formulas : 
n == sin 2 i, n — sin i, and n — i/ sin i, 
n being the percentage of bodies whose incidence angle is less than i. 
The graphic representation of n in each case is the area beneath the curve 
from the vertical axis to the ordinate corresponding to i. The curves 
themselves represent the differential equations: 
wrt ck • 
—rr = 2 sm i cos i, 
di 
dn 
di 
= cos i, 
and 
dn _ cos i 
di 2 j/ s i n { 
in which dn is the relative number of bodies having the incidence 
angle i. 
