392 On the Variation of the Moon’s Motion. 
acquired or lost by the moon while passing through an octant, 
which must be the amount by which its velocity exceeds the 
mean in syzygy, or falls short in quadrature. The mean rate of 
the moon’s motion in its orbit being about 200224 feet per min- 
ute, the velocity in syzygy must exceed the mean by about 35%y°s'sq 
or ;i;, and that in quadrature fall short by the same fraction. 
Consequently, so far as the cause we have now considered is con- 
cerned, the numbers 221, 222, and 223 are proportional to the 
velocity in quadrature, the mean velocity, and the velocity in 
syzygy- , 
We have thus far supposed the moon’s orbit to retain its cir- 
cular form, notwithstanding the disturbing influence of the sun; 
but this is not possible. 'To retain a body ina circular orbit, the 
centripetal and centrifugal forces must be equal. But we have 
just shown that the velocity in syzygy is greater than in quadra- 
ture; and as the centrifugal force is proportional to the square of 
the velocity, it must also be greater. On the contrary the ex- 
3EF? 
pression for the ablatitious force, viz. ———___. 
when the moon is in quadrature at A or B, but is increased when — 
-8ME?.. _3ME.. 
expression becomes + cians 
of the terms, that this is just double the maximum tangential 
force, and as the numerical value of that, expressed in decimals of 
the moon’s mean gravity towards the earth, was 008306625, this 
must be 01661325, or about ;';. Consequently the moon’s grav- 
ity toward the earth at A or B is to its gravity at C or D in the 
ratio 60 : 59. 
The force of gravity being thus diminished, and the velocity 
increased when the moon is in syzygy, the centrifugal force ex 
ceeds the centripetal, which causes the moon’s track to fall with- 
out the circle, asto F and G, Fig. 2 
(fig 2.) But when the moon G 
is in quadrature the cenfripe- 
tal force exceeds the centrifu- ® 
gal, causing it to fall within its ® 
circular orbit as to L and N. | Pg 
m, becomesO 
It will be seen by a comparison 
: fe wai eee 
bee OR ee 
eRe eS See ee 
