31 
1920-21.] The Equations of Motion of a Single Particle. 
which lead to results different from those just obtained.* It will be seen 
below that in certain important cases, in addition to the case discussed 
above, a change of variable enables us to bring (11) into the form 
2 (a-V)dyl , 
where dsl depends only on x 1: x 2 , x 3 , and dyjdty 4 = 1 . Comparing this form 
with ds 2 0 — c 2 dt 2 , it is natural to interpret y 4 as t and to regard 2{a — Y) as 
the square of a variable velocity of light. From this point of view, then, 
we should expect V2(<x — V) to satisfy the generalised Laplace equation 
rather than V itself, especially as when a is large the value of V so 
determined will agree with the Newtonian potential to a first approxima- 
tion. In the case discussed in the preceding paragraph this second method 
of determining V leads without approximation to the Newtonian form. 
§ 5. From the dynamical point of view, there does not appear to be any 
reason for imposing any particular form of restriction on V and A in addi- 
tion to the general restrictions of § 1 and § 2, which are sufficient to ensure 
that the equations of motions are equivalent to the Newtonian equations 
to a first approximation. In view of the particular case partially developed 
in § 3, it is, however, natural to investigate the case in which A depends on 
V alone except for an additive term of the form — ax 4 (a constant) introduced 
in the integration of (8). This makes the equations of the system depend 
solely on the potential. We shall not impose this condition in its entirety 
at first, but will introduce it as required in the course of the discussion. 
We shall first determine conditions under which it is possible so to 
transform the variables as to remove the term dkdx± in (11). Setting 
Vi — (i= 1, 2, 3), 2/4 = dfai) *^ 3 J ‘*' 4 )’ 
Xi - Vi (i=l, 2, 3), X^=f(y v y 2 , y s , y 4 ), 
in (11), and equating the coefficient of dy^dy^ to zero, we get 
( 21 ) 
dg 0A 
chq dx^ 
2 ( v+a 4>S,. 
=0 (*“ 1, 2, 3). 
If A is a function of V and x } alone and V is independent of ,x 4 , this leads to 
( 22 ) 
0 A 3 A_ 2 /^ +y )Y=o, 
dx, 0Y \dx L 
0V 
any solution of which will give the required result. If 0 A/ 0 V and 0A/0x 4 
are functions of V alone, f i.e. A= — acc 4 +/(V), the solution of (22) is 
dV 
(23) 
2/ 4 = F (* 4 + j |^ 
av (V-a 
* Gf. Levi-Civita, “Statica Einsteiniana,” Rom. Acc. L. Rend., xxvi (1917), p. 458, where 
the connection with Einstein’s equations is discussed. 
+ There are, of course, other cases in which an explicit solution of (21) can be obtained, 
e.g. when 2Y + A is a function of A and x i alone, or if A has the form AfiY) + x 4 A 2 ( Y). 
