26 
Proceedings of the Royal Society of Edinburgh. [Sess. 
II —On the Equations of Motion of a Single Particle. 
By J. H. M. Wedderburn. 
(MS. received August 9, 1920. Read November 1, 1920.) 
§ 1. When solved for the second derivatives, the Lagrangian equations of 
motion for a system in which there are no extraneous forces have the form * 
d 2 x k ^ ( i j \ dxidx 1 _ 
dt 2 ^ ) i j dt dt ~ ’ h- 
ij ’ 
h 2, . . . tz), 
or, disregarding the parameter t , 
(1) . . <^+z{y (*=i,2 
ij 
| Y J- being the second Christoffel symbol j* of the matrix associated with 
the kinetic energy. If there are extraneous forces and, denoting t by 
x n+1 , we add d?x n+1 — 0 to the set of equations, the equations of motion are 
(2) d 2 x k + 2 | Y \dxidXj-¥ k dx n + 2 = 0, (k= 1, 2, . . . n), d 2 x n+1 = 0. 
ij ' 
These equations have a general similarity to (1) with the number of 
variables increased by one, and would in fact have exactly the same 
mathematical form if there existed a matrix \js for which 
(3) 
u i “Ur 
\i 7 
U 
+i y=o, 
fn + 1 7 
\ i 
\'nAi 
o, (i,j= b 
= -Ffc, (i,j,7c=l, 2, . . . n), 
71 + 1), 
where the dashes indicate symbols belonging to the new matrix \js. It is 
readily seen, however, that these relations cannot hold in general, and it is 
the principal object of this note to investigate the circumstances under which 
they can be approximately satisfied. If ordinary dynamics can be modified 
so that this is so, and the time f variable is regarded as a special case of the 
space variables, it will follow that the equations of motion in the modified 
* Of. Whittaker, Analytical Dynamics , p. 39 ; or Wright, Invariants of Quadratic Differ- 
ential Forms , p. 83. 
t The definition and properties of these symbols may be found in Wright, loc. cit., p. 10. 
I If there is more than one particle in question, it may be necessary to introduce more 
than one time variable. In many ways it is best to consider these as strictly space variables 
and. to assume that particles in our universe are moving in the direction of the fourth space 
direction so nearly uniformly in straight lines that the distances measured in this direction 
are, to a first approximation, proportional to the time. 
