28 
Proceedings of the Royal Society of Edinburgh. [Sess. 
( 8 ) 
If a. 
a --i) <* =1 ’ 2 ’ 3 >- 
Finally, comparing coefficients of dx\ in (6) and using the relations just 
derived, we have 
0Y da k . . da.. d 
• —= 
dx k dx 4 z dx k dx k 
a 22 , a 33 are all different, we infer that Y has the form 
9 i( x i)+ 92 ( x 2 )+ 9 s( x 3 )’ or if say a n = ‘W a 33> it has the form g 1 (x v x 2 )+g s (x 3 ) ; 
but since the x’s are ordinary rectangular coordinates, this form does not 
agree to a first approximation with the Newtonian potential, so that when 
we are dealing with Newtonian dynamics we must have a u = a 22 = a 33 and, 
being constant, we may set each equal to unity. This gives immediately 
0A 
(9) 
a, , = 2 Y 4- 2 ; 
di 
the arbitrary function of x 4 introduced by the integration being included 
in A. The fourth equation of (5') then becomes 
dxidx. 
* V - 1>. - - i - 2 i(S ■ *££) 
so that in order that d 2 x 4 may be a second order quantity it is sufficient 
to assume that Y + 0A /dx 4 is large compared with the other coefficients in 
the equation, a condition that can be readily attained, since V + 0A/0as 4 
contains an arbitrary function of x 4 which can have terms not entering 
into any other coefficient. 
We assume, therefore, that \js has the form 
10 0 A, 
( 10 ) 
Ac 
A 3 2 Y + 2A 4 
( 12 ) 
where we have set A { for dA/dx t . The corresponding quadratic form is 
(11) . . . . . 2 dx\ + 2 Y dx\ + 2dAdx 4 , 
and the corresponding Lagrangian equations can be readily reduced to 
d 2 x k + A k d 2 x 4 - Y k dx\ =0 [k = 1, 2, 3), 
(2Y + A 4 )d 2 x 4 + d 2 A + 2 dVdx 4 - Y 4 dx\ = 0, 
the form (11) equated to a constant being, as is well known, a first integral 
of (12).* 
* This seems the most natural way of introducing the form (11) into the dynamical 
system. Jt suggests that it is natural to consider this form as defining time rather than 
distance, thus leaving open the possibility of using a different form to define the geometry 
of space. 
f dh 
■ |(2f 
