LECTURE X. 



THE ANNUAL EQUATION. 



LET us next take into account the effect of the first power of the 

 eccentricity of the Earth's orbit. We shall find that it produces an ine- 

 quality in the Moon's coordinates, the chief part of which has a period 

 of one year, and is therefore called the Annual Equation. 



In the formulae of Lecture VII, let the known approximate solutions 

 l a , , include the Variation only ; then the equations for the corrections 

 81, 80 are 



X + ~ + 4o. sin 2i/ - 3 4 (1 + 3o. cos 2ii) 



at- r at JV 



- 2 [( 1 +n') + 26, cos 2</r] ^- + 3n'' 2 (sin 2$ + b, sin 4/) 80 = 0, 



72S<3 fl?\fi 



Y+ - + 4a 2 sin 2// - + Bn 1 " (-b, + cos 2^ + 6 2 cos 4i/) 86 



7C\7 



-^ =0, 



where a,, 6 2 , /t/a s are known quantities whose values are given in 

 Lectures IV, V. 



Refer now to Lecture III, and we find that the terms that are left 

 outstanding when the terms of the Variation are substituted, and the 

 parallactic terms omitted are the following : 



o 01 



X - - nV cos (n't - O - = n'V cos {2 (6 - n't) - (n't - m')} 



Y= + ^ nV sin {2 (^ - n't) - (n't - /)} 



nV sin {2 (^ - n't) + (n't - /)} 



A. II. 



