518 



turhing 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 : 



^ = K2/3 + 3/3-g/3/3)+^ (/3 + 3y->/37) ; (35) 



in which the laws of the circular revolutions of the vectors j3 

 and y give 



f=-P; 5=_»,V (36) 



Assuming, from some general indications of this theory, 

 an expression for the perturbation of the moon's vector, which 

 shall be of the form 



8i3=m^(^j3 + 5y-'j37+q3-'7-'/37i3), (37) 



and neglecting all powers of m above the square, we find 



^= -mV/3 + 57-^/3y + 3-^C/3-V'^7^); (38) 



i3-'Si3.i3 = m\A^ + Cy-'^y + ^^-'7- '^7^) ; (39) 



so that the three numerical coeflBcients, A, B, C, must satisfy 



the three following equations of condition : 



-A = 2A + li giving ^=r -I ; (40) 



and 



-B=i(5 + 3C) + |; -9C=l(C+3B); (41) 



