29] DUE TO SECULAR CHANGE OF THE PLANE OF THE ECLIPTIC. 251 



Hence, by substituting for k sin (it + e), k cos (it + e), &c. in Laplace's 

 expression for s,, their values in terms of p, q and their differential co- 

 efficients, we find 



[1 dp 1 d-q 

 __ ___^ __A_ __ 

 m 2 dt (f m 2 ) 2 de 



f 1 dq I 



\ o ~. 



[_ |m- c 



+ cosv 



which exactly agrees with Hansen's expression in his Darlegung, p. 490*, 

 except that Hansen's argument f+ u> 9, represents the longitude on the 

 orbit, whereas Laplace's argument v is the longitude on the ecliptic ; but 

 these two longitudes may be employed indifferently in terms of the order 

 of small quantities to which the approximation is restricted. 



Laplace remarks that -m" is at least 4,000 times greater than 2i, and 



he therefore infers that the above value of s 1 may be neglected as in- 

 sensible. If, however, the numerical values of the quantities denoted by k 

 had been known to Laplace, he would have seen that some of those values 

 are very considerable, exceeding one degree, and therefore that j^-^ f this 

 amount is by no means to be neglected. 



Finally, we will reduce Laplace's transformed expression to a form 

 immediately comparable with our former results. 



The velocity perpendicular to the ecliptic of a point in any arbitrary 

 longitude L is represented in one system by 



dp T do . 

 f- cos L + jj sm L, 

 dt dt 



and in the other system by 



wsin(-a). 



Hence = CD sin C, 



* In this expression - is equivalent to 6 + b't in Hansen, and ^ is equivalent to c + c't. 

 at dt 



Also Hansen's expression n(a + rj), which denotes the mean motion of the Moon's node, 

 is equivalent to f m? in Laplace, as the latter takes n, the Moon's mean motion, to be equal 

 to unity. 



322 



