851 
The Varianonal Displacements. 
2. We consider a field of streaming discrete particles, the velo- 
cities being continuous funetions of the coordinates and the time. 
We imagine a picture in a four-dimensional space-time-extension 
designing the motion-trails of the particles indicating their positions 
in successive instants. Now the displacements will consist of a shift 
in space and a shift in time, and we shall define these shifts with 
the aid of a field of a four-fold vector 7“, the components of which: 
rr, 13), being space-components and 7@) being the time-compo- 
nent, will be continuous functions of the coordinates and time «4 
(ad . 4). 
Mathematically, we define the shifts as the one-membered infini- 
tesimal transformation group determined by the functions r¢ (a = 1.. 4), 
with parameter 0: 
or 
4 ct 
Agard 40S (ret. 
1 Ox 
This will be clearer if we explain the nature of the r¢. If the 
variational parameter increases by an amount d@, then the particles 
are supposed as suffering an additional shift given by 
72 dO (a = 1, 2, 8, 4), 
the values of +7 being taken such as they are in the momentary 
point-instant occupied by the particle. Leaving out second order 
terms with 6%, we at once see that the first approximation of the 
total shift will be 
6 re, (a =de, 4), 
and proceeding to second order terms we obviously get 
6 
A ne — Í 
« 
0 
Ara 
A at = Ora + ws = (Oe 6 
ae 
Ore 
ra {+ > (c) 3 — 7c vd dy, 
axe 
where now the values of 7 and their derivatives have been taken 
in the point-instants of the particle’s undisturbed motion. 
The Variation of the Stream. 
3. The following conception of the stream components will greatly 
facilitate our deductions. 
Let N, a continuous function of space-time-coordinates, represent 
the density of the particles’ distribution through space. At the instant 
