LECT. XV.] MOTION IN AN ORBIT OF ANY INCLINATION. 



tT Zdt 



69 



Now 



-f^j- = -, 

 Tm v 



where v is the resolved part of the velocity at P perpendicular to the 

 radius vector. But 



H = vr ; 



hence the angle through which the orbit is turned in an indefinitely short 

 time dt is 



rZ , 



To find the corresponding changes in the elements that determine the plane 

 of the orbit, namely, the inclination of the orbit to a fixed plane, and the 

 longitude of the node on that plane. Let NPQ be the great circle which 



represents the plane of the orbit at the time t, NR the plane of reference, 

 usually the plane of the ecliptic, P the position of the body at the same 

 time, and let i = PNR, the inclination, and let JV be the longitude of the 

 node. 



Let nPq be the position of the orbit at time t + dt. 



Take NQ 90 ; draw nm perpendicular to NPQ and qQR perpendicular 

 to NR; then QR = i; and by what we have just proved 



Therefore 



where 

 But 



JV V 



nm = YT dt . sin 0, qQ = ^fdt. cos 6, 



1 ti 



6=nP. 



nm = Nn sin i = sin i dN ; qQ = di. 



