SECT. 2.] TO POSITION IN SPACE. 61 



cos (N-- Q, ) a sin A 

 cos i sin (N & ) = a cos A 

 sin (N 8 ) = b sin B 

 cos a cos (^V 8 ) = & cos 5, 



(see article 14, II.). We shall then evidently have 



x = ra sin (u -\- A) 

 y = r b sin (u -j- .Z?) 

 3 = r sin sin M . 



54. 



The relations of the motion to the ecliptic explained in the preceding article, 

 will evidently hold equally good, even if some other plane should be substituted 

 for the ecliptic, provided, only, the position of the plane of the orbit in respect 

 to this plane be known ; but in this case the expressions longitude and latitude 

 must be suppressed. The problem, therefore, presents itself: From the known 

 position of the plane of the orbit and of another new plane in respect to the ecliptic, to 

 derive the position of the plane of the orbit in respect to the new plane. Let n Q , Q Q , 

 n be parts of the great circles which the plane of the ecliptic, the plane of the 

 orbit, and the new plane, project upon the celestial sphere, (fig. 2). In order 

 that it may be possible to assign, without ambiguity, the inclination of the second 

 circle to the third, and the place of the ascending node, one direction or the other 

 must be chosen in the third circle, analogous, as it were, to that in the ecliptic 

 which is in the order of the signs; let this direction in our figure be from n toward 

 Q . Moreover, of the two hemispheres, separated by the circle n& , it will be 

 necessary to regard one as analogous to the northern hemisphere, the other to 

 the southern ; these hemispheres, in fact, are already distinct in themselves, since 

 that is always regarded as the northern, which is on the right hand to one moving 

 forward* in the circle according to the order of the signs. In our figure, then, Q, 

 w, & , are the ascending nodes of the second circle upon the first, the third upon 

 the first, the second upon the third; 180-- n Q, Q , &n& ,nQ, Q, the inclina- 



* In the inner surface, that 5s to say, of the sphere represented by our figure. 



