OF THE MOTION OF A SYSTEM. 213 



motion of the bodies bears no determinate relation to x, y, 

 and z ; and, if the attractive force vanishes, the principle 

 is also true with respect to any point whatever : nor is the 

 demonstration limited to changes produced by insensible 

 degrees. 



324. Lemma. If we have two systems 

 of orthogonal coordinates, j?, ?/, z^ and ^V//? ^///> 

 2f,,„ originating from the same point : if e be 

 the incUnation of the plane of ^,,, and t/^^^ to 

 that of a: and ?/, [its positive values implying 

 that z^^, inclines towards the same side of or 

 with +?/], and if ^ be the angular distance of 

 ^ from the intersection of these planes, and (p 

 that of .r,,„ the equations between the coor- 

 dinates will be 



a:=a^„, (cos Q sin ^ sin ^+ cos v^ cos (p) 



-\-y,„ (cos B sin ^ cos <p — cos ^ sin <p) 



-\-z,„^m 6 sin ^^ 

 y=x„, (cos 5 cos ^^ sin ^ — : sin A^ cos ^) 



+^/// (cos d cos -^ cos ^+ sin 4. sin ^) 



+z,„ sin 5 cos ^^ 



2:=:2:„, cos ^ — ^7/,,, sin Q cos <p — 0?,,, sin 5 sin ?». 



In order to assist the imagination, we may suppose the 

 origin of the coordinates to be at the centre of the earth, 

 the plane of x and y to be the ecliptic, and z to be directed 

 to its north pole [x being considered as positive when it 

 tends more or less to approach the vernal equinox op, and 

 y when it tends towards the sign Zo, and negative on the 



