29] DUE TO SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. 249 



Also since the Moon's node takes about 18 '6 years to perform a com- 

 plete revolution 



. 27T , 



c cos i = c nearly. 

 lob 



w 0-479x18-6 



Hence . . // = - . expressed m seconds. 



c cos^ sin 1 2vr 



which agrees with the value of the coefficient of the principal term found 

 in the former investigation. 



The form above found for 8ft suggests a very simple geometrical inter- 

 pretation of this inequality in latitude. 



If we suppose a fictitious ecliptic to be inclined to the true ecliptic 



at the angle l"'42, the circular measure of which is - . , and if we also 



c cos i 



suppose that the longitude of its ascending node on the true ecliptic is 

 90 + (7, then the elevation of the fictitious above the true ecliptic cor- 

 responding to the longitude 6 will be 



&> 



-.sin (0-90 + (7), 

 c cos^ 



.cos(0-(7), 



C COS I 



Hence the latitude above the fictitious ecliptic will be equal to ft, that 

 is, the expression for the Moon's latitude with respect to the fictitious 

 ecliptic is the same as the expression found for the latitude in the case 

 when the ecliptic is taken to be a fixed plane. 



This geometrical interpretation of the inequality was first given by 

 Hansen. 



III. Note on the Mecanique Celeste, tome m. p. 185 (edition of 1802). 



At any arbitrary point whose longitude is \, Laplace takes the elevation 

 of the variable ecliptic above the fixed plane of reference to be represented by 



A. 



32 



