520 
second fictitious moon, being so placed that the actual moon 
is midway in the heavens between ¢his fictitious moon and the 
one which was before considered. These two latter terms of 
(43) contain the chief laws of the Lunar Variation ; and are 
easily shown to give the known terms in the expressions of 
the moon’s parallax and longitude, : 
1 11m? 
d8-=m’cos2()—©); d= 
r 8 
It may assist some readers to observe here, that when the in- 
clination of the orbit is neglected, the longitudes of the first 
and second fictitious moons are, respectively, 
2©—), and 3)—20; (46) 
while those of the first and second fictitious suns, mentioned 
in a former section of this abstract, are, under the same con- 
dition, 
sin2()—@). (44) 
2) —O, and 3@—2). (47) 
VII. The law and quantity of the regression of the Moon’s 
Node may also be calculated on principles of the kind above 
stated, but we must content ourselves with writing here the 
formula for the angular velocity of a planet’s node generally, 
considered as depending on the variable vector of position a, 
2 
the vector of velocity s, and the vector of acceleration = 
and also on a vector unit A, supposed to be directed towards 
the north pole of a fixed ecliptic. The formula thus referred 
to is the following : 
-aAd.s.d@?adaa 
dg = So heiatGee ¢ 
(v.AV.ada)? 
where s and vy are, as before, the characteristics of the opera- 
tions of taking the scalar and vector of a quaternion. The 
author proposes to give a fuller account of his investigations 
on this class of dynamical questions, when the Third Series 
of his Researches respecting Quaternions shall come to be 
printed in the Transactions of the Academy: the Second Se- 
(48) 
