SECT. 4.] PLACES IN SPACE. 155 



111. 



Let us suppose, in the second place, the two places to be given by means of 

 their distances from three planes, cutting each other at right angles in the sun ; 

 let us denote these distances, for the first place, by x, y, z, for the second, by 

 x, i/ , z , and let us suppose the third plane to be the ecliptic itself, also the posi 

 tive poles of the first and second planes to be situated in N, and 90 -j- N. We 

 shall thus have by article 53, the two radii vectores being denoted by r, /, 



x = r cos u cos (N 8 ) -f- r sin u sin (JV Q ) cos i, 



y = r sin u cos (N Q, ) cos i r cos u sin (N Q ) , 



z = r sin u sin i 



x = r cos 11 cos (N 0,}-\-r sin u sin (N & ) cos i, 

 y = / sin u cos (N 8 ) cos i / cos ut sin (N Q ), 



z r sin u sin i, 

 Hence it follows that 



zy yz = rr sin ( ) sin (N Q, ) sin i, 

 xz zx = rr sin (u ) cos ( JV Q ) sin i, 

 xy 1 yx = rr sin (u r u) cos i. 



From the combination of the first formula with the second will be obtained JV & 

 and r r sin (u f u) sin i, hence and from the third formula, i and rr sin (u u) 

 will be obtained. 



Since the place to which the coordinates x , y 1 , z , correspond, is supposed pos 

 terior in time, u must be greater than u : if, moreover, it is known whether the 

 angle between the first and second place described about the sun is less or greater 

 than two right angles, rr sm(u w)sinz and rr sin(u u} must be positive 

 quantities in the first case, negative in the second : then, accordingly, N 2 

 is determined without doubt, and at the same time it is settled by the sign of 

 the quantity xy yx , whether the motion is direct or retrograde. On the othei 

 hand, if the direction of the motion is known, it will be possible to decide from 

 the sign of the quantity xy y x , whether u u is to be taken less or greater 

 than 180. But if the direction of the motion, and the nature of the angle 



