60 RELATIONS PERTAINING SIMPLY [BOOK I. 



53. 



The position of any point whatever in space is most conveniently deter 

 mined by means of its distances from three planes cutting each other at right 

 angles. Assuming the plane of the ecliptic to be one of these planes, and denot 

 ing the distance of the heavenly body from this plane by z, taken positively on 

 the north side, negatively on the south, we shall evidently have s = r tan ft = 

 r sin ft = r sin i sin u. The two remaining planes, which we also shall consider 

 drawn through the sun, will project great circles upon the celestial sphere, which 

 will cut the ecliptic at right angles, and the poles of which, therefore, will lie in 

 the ecliptic, and will be at the distance of 90 from each other. We call that pole 

 of each plane, lying on the side from which the positive distances are counted, 

 the positive pole. Let, accordingly, N and N -\- 90 be the longitudes of the 

 positive poles, and let distances from the planes to which they respectively 

 belong be denoted by x and y. Then it will be readily perceived that we have 



a; = r cos(X N} 



= r cos ft cos (X 8 ) cos (N 0,}-\-r cos ft sin (X Q ) sin (^V Q, ) 

 ^ = /sin(Jl N) 



= r cos ft sin (X Q ) cos (JV Q, ) r cos ft cos (X Q ) sin (N Q ), 



which values are transformed into 



x = r cos (N 8 ) cos u -\- r cos i sin (N & ) sin u 

 y = r cosz cos (N 8) sin u rsin (N 8) COSM. 



If now the positive pole of the plane of x is placed in the ascending node, so that 

 N= 8, we shall have the most simple expressions of the coordinates x,y, z, 



x = r cos u 



y =. r cos i sin u 

 z =. r sin i sin u . 



But, if this supposed condition does not occur, the formulas given above will 

 still acquire a form almost equally convenient, by the introduction of four 

 auxiliary quantities, a, I, A, B, so determined as to have 



