29] DUE TO SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. 245 



Hence by what has been before said if N', i' denote the longitude 

 of the node and the inclination at the time t, with respect to the variable 



dN' di' 



ecliptic, , and -j- will be given by the following formulae : 

 dt dt 



' 



= - c cos i' + F(6, 8'} + a. sin (N' - C) cot i', 



~ 



in which F(6, 6'}, f(9, 9'} now involve the elements N' and i', instead of 

 N and i. 



Now let N be the longitude of the node, and i the inclination at 

 the time t, on the supposition that the ecliptic remains fixed, all the other 

 circumstances of the Moon's motion remaining unaltered ; then we have as 

 before 



, ff), 



ar 



Let 

 and i' = i + Si, 



where SN and Si are entirely due to the motion of the ecliptic and there- 

 fore vanish with <u*. 



Then neglecting the square of and supposing the value of 0, or the 

 Moon's longitude, to remain unchanged, we have 



dSN ... fdF\m , fdF\ s . , . ,,, n . .. 



r = c sin 181 + f-Tjrd SN + (,.-) di + o) sin (N- C) cot i, 



**r * /-\T n\ 



= - }8N+ ( j. Si &> cos(^-C). 



dt \dNJ \di] 



(dF\ IdF 

 Now ' (di 



and 



(df\ (df\ 



\dN]' \di/' 



* It is hardly necessary to mention that 8./V and Si are here employed in a wholly 

 different sense from that in which the same symbols were used, for a temporary purpose, 

 in the earlier part of this investigation. 



