On the Variation of the Moon’s Motion. 391 
When MED =45°, and consequently 2MED= 90°, its.sine be- 
comes EB, and the expression for the tangential force becomes 
a Or by giving to m and SM their mean values it be- 
comes iin 4 8 EBs asx EB . 
: SE SE? SE: 
Restoring the numerical value of 8, viz. 354936, and giving 
to EB and SE their mean values, viz. EB=237577 miles, and 
SE = 95024608 miles, the value of the expression last found be- 
comes -008306625. Its mean value through each octant must 
be proportional to the mean sine in a quadrant, which is the sine 
of 39° 28’ nearly =*63563. Hence, 1 : 63563: : 0083066 : the 
mean tangential foree in an octant or a quadrant, which is thus 
found to be ‘00528. 
Increments or decrements of velocity are proportional to the 
forces that produce them, and consequently the total velocity ac- 
quired or lost is proportional to the sum of the forces, whether 
they be uniform or variable in their operation. Hence the amount 
which the moon gains or loses in velocity while passing through 
a quadrant, is precisely the same as it would be if the tangential 
foree remained all the while in its mean state; and to see more 
clearly what this amount is, let us compare the tangential force 
with some other, with whose accelerating power we are familiar. 
The force of gravity at the earth’s surface, we know, draws a 
body 16,1, feet per second, or 57,900 feet per minute, thus crea- 
ting in one minute a velocity of double that amount, or 115800 
feet per minute. But as the moon is about 60,', times’as far 
from the center of the earth as the earth’s surface is, the dimin- 
ished force of gravity at that distance would give a velocity only 
about 5;';5 as great, or about 32} feet per minute. ‘The mean 
tangential force being ‘00528 of the earth’s attraction on the moon, 
will create a velocity of only this fraction of 32; feet per minute, 
which is equal to about 2,'; inches. Multiply this by 53155, the 
number of minutes it takes the moon to describe a synodical oc- 
tant,* and we obtain about 904 feet per minute, as the velocity 
EE en) 2 wT se re 
* The term octant is not strictly true, for by reason of the earth’s progressive 
ion in its orbit, the moon describes an are of 97° 16/ 35/’ in passing from syzygy 
to quadrature or from quadrature to syzygy, and consequently there is 48° 38’ 174” 
in the are which we have called an octant. We employ the term for the sake of 
conciseness. . 
