1920-21.] The Equations of Motion of a Single Particle. 29 
When the potential is independent of the time x 4 , and dA/dx 4 is constant, 
equal to —a, say, a first integral of the last equation of (12) is 
(2V - a)dx± + dA = const., 
or 
2(Y - a)dx 4 + 2 t^dx^ = const. 
§ 3. An important case of the equations of the preceding paragraph 
arises when (i) the potential Y is a function of r alone, and (ii) A= — ax i} so 
that A 4 = — cl, & constant. 
When (ii) holds and also V 4 = 0, equations (11) and (12) become, on 
introducing a parameter r, 
(13) . 
(14) . 
whence 
(14') . 
dxA 2 
da 
(H constant) 
d 2 x$ 
~d 7- 
Y A 
( dx. 
\dr J 
V- 
0, (*=1,2,3) 
2(V - a)^i + 2~ d p = 0, 
dr 2 dr dr 
— £ = ------ ( K constant) 
dr Y — a 
so that, setting U = — k 2 I( V — a), 
(15) . 
(15') . 
d 2 x^ 
d ? 
U* (*=1,2,3), 
dx^ U 
dr k 
2U = H. 
If now V, and therefore U, is a function of r alone, it follows by the 
usual methods that the motion is in a plane, and, choosing spherical polar 
co-ordinates so that this plane is <p = 0 , the equations of motion become 
d?r _ (d,6\ 2 _ k 2 dY = dU ' 2 d0 = j 
dr 2 \dr) (V - a) 2 dr dr’ dr 
where h is constant, or setting u—l/r, 
(16) . 
dU 
d 2 u _ _ dr _ 1 dU dO _ , 2 
’ dO' 2 h 2 u 2 h 2 du dr 
If we now assume that Y has the Newtonian form m/r, and set /c = — a so 
that equals r when r* is large, then the equation for n becomes on 
neglecting terms in 1/a 2 , 
(16') . 
d 2 u 
dO 2 
5 + 
~Wa 
m 
h 2 ' 
