29] DUE TO SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. 241 



Also s being supposed very small, 8s is equal to the circular measure 

 of the change of the Moon's latitude due to the secular change in the 

 plane of the ecliptic, and if we divide 8s by sin 1" we shall find the change 

 of the latitude in seconds 



cos (-nt + 2n'i) -231 '0661 cosnt 



- 7 

 siri i 



- 1-1789 cos (3nt - 2n'i) - 0'0079 cos (5nt - 4n't)}. 



Now, according to the data adopted by the Astronomer Royal, the 

 circular measure of the angular motion of the plane of the ecliptic in 1 

 year is 0'479 sin I". 



Also 1 year is represented in our notation by the time . 



n 



_ 

 Hence o> = 0'479 sin 1", 



it 



co . n' 



and -^-// = 0-479 ~ = 0'OOG16,354. 



sm 1 2vr 



Therefore the inequality of latitude expressed in seconds is 



0""0002 cos ( - 3nt + kn't) + 0"'0485 cos (-nt + 2n't) - l"'4242 cos nt 



- 0"D073 cos (3nt - 2n't). 



In this expression the mean longitudes nt and n't are reckoned from 

 the node of the variable ecliptic upon the fixed ecliptic. If the mean 

 longitudes are reckoned from the equinox in the ordinary way, and if C 

 be the longitude of the above-mentioned node, we must replace nt and n't 

 in the above by nt C and n't C respectively, and the expression for the 

 inequality in latitude becomes 



0"'0002 cos ( - 3nt + 4n'< - C) + 0""0485 cos (-nt + 2n't - C) 



- 1"'4242 cos (nt -C)- 0"'0073 cos (3w - 2n't - C). 



In the above investigation the quantities co and C are supposed to 

 be constant. If these be subject to small secular variations, the differential 

 equations become a little less simple, but are easily formed, and the above 

 solution will require the following modifications, viz. 



(1) Instead of the constant value of co we must employ the variable 

 value which is of the form 



co + a>'t; 



31 



