518 
turbing force, and occasions the presence of the first of the 
two parts of the second member of the equation (34): which 
equation will be found to contain a considerable portion of 
the theory of the moon. 
VI. The author will only mention here two very simple 
applications, which he has made of this equation (34), one to 
the Lunar Variation, and the other to the Regression of the 
Node. ‘Treating here the sun’s relative orbit as exactly circu- 
lar, and the moon’s as approximately such, neglecting the in- 
clination, taking for units of their kinds the sum of the masses 
of the earth and moon, and the moon’s mean distance and 
mean angular velocity, and employing, as usual, the letter m 
to denote (not now the earth’s mass, but) the ratio of the sun’s 
mean angular motion to the corresponding motion of the moon, 
the differential equation (34) becomes : 
Gee = 3B +38-8B B)+ B+3y"B)s (85) 
in which the laws of the circular revolutions of the vectors 3 
and y give 
Assuming, from some general indications of this theory, 
an expression for the perturbation of the moon’s vector, which 
shall be of the form 
éB=m*( AB + By" By + CP~'y~ ByB), (37) 
and neglecting all powers of m above the square, we find 
TeP  —m(4B + ByBy +3°CB'y"ByB)s (38) 
B~08.B =m*( AB + Cy'By + BB y"'ByB); (39) 
so that the three numerical coefficients, 4, B, C, must satisfy 
the three following equations of condition : 
—A=2A+3; _ giving d=—}; (40) 
and : . 
—B=}(B+3C)+3; —9C=3(C+3B); (41) 
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