tee oe 
On the Variation of the Moon’s Motion. 393 
‘The effect is to throw the Fig. 3. A 
orbit into something such a hipsioge 
shape as is represented in fig. , D 
3, viz. a kind of oval, with its A 
longest diameter, AB, at right _ ' 
angles to the line ES, drawn ' B | 
from the earth to the sun. 
By this change both the moon’s gravity toward the earth, and. 
its velocity, are still farther affected ; its gravity because its dis- 
tance from the earth being not now the same in all parts of its 
orbit, the attractive force of the latter varies in the inverse ratio 
of the square of the distance ; and its velocity because the direc- 
tion of its motion being rendered oblique to the radius vector, the 
earth’s attraction conspires with the tangential force of the sun 
to accelerate it while passing from quadrature to syzygy, and re- 
tard it while passing from syzygy to quadrature. Indeed so far 
as the attractive force of the earth is concerned, being directed 
toward a fixed point, the moon’ describes equal areas in equal 
times about the earth as a centre, and consequently the velocity 
at C or D : the velocity at A or B::EA : ED. 
There are two reasons then why the velocity of the moon is 
greater in syzygy than in quadrature ; Ist, the tangential force of 
the sun, which increases it in the ratio 221 : 223; and 2d, the ob- 
lique action of the earth’s attraction on the moon in its disturbed 
orbit, which increases it in the ratio ED: EA. Hence from both 
combined it is increased in the ratio 221 x ED ; 223 x EA. 
There are also two causes which affect the moon’s gravity 
toward the earth; Ist, the ablatitious force of the sun, which 
makes it less in syzygy than in quadrature in the ratio 60 : 59; 
and 2d, the unequal attraction of the earth in the disturbed orbit, 
which makes it greater in the ratio ED? : AE*. Hence from 
both combined, the gravity in quadrature : the gravity in sy- 
zygy::60x ED? : 59x EA’. 
A near approximation to the shape of the oval, or to the ratio 
between its least and greatest diameters, can be determined with- 
out much difficulty. ie : 
It is a law of central forces that the radii of orbits are propor- 
tional to the square of the velocity divided by the centripetal 
force. Hence the moon’s orbit at C and D must be described 
(223 x EA)? , 
59xEA? - 
with a longer radius than at A and B, in the ratio 
