, 
oo 
a wip? “a? yx? ad 
§ — y= —— | — dr + — fa’dr. 
el r° dl 
r 0 
But whatever may be the particular properties of the central 
body, we can put P= Q=o=0 and a=a, at a large distance, 
in consequence of which we find from (15) 
xa. a 3B 
5 — 2 is —— 
ers en 3 : Sy hae os pee 
8c? 7? 
in which B is a constant of the second order. 
From this it follows that 
x a 2B B 
p=—l + — + OSS a eee: 
8 c? r? r? r 
6. We shall now examine how a particle moves in the field of 
a single centre. 
The motion is determined by a principle corresponding to that 
of HAMILTON, viz. 
to to 
sf Lam deM ge tel verver gnd. 
ty t 
In the case under consideration, we have 
ds? = v (da? + dy? + de?) + (u—v) dr? + wdt’. 
If we introduce polar coordinates r, 3, p, we get 
ds* = wdt? + udr* + vr? dĲ* + vr’ sin? H dy’, 
hence 
L=Vw es ur? + or? B? +L cr? sin? 9 p°. 
One of the three equations of motion is 
a(aL\_, 
ase 
which shows that if ¢ once is zero, it remains so; we see from this 
that the motion takes place in a plane, and, knowing this, we can 
~ 
choose the coordinates so that this plane becomes the plane 9 ==. 
md 
Accordingly 
L=Vw a ur? + or g? 
and the equations of motion become: 
d 
dt Or Or dt Òp 5 
The equation of energy 
