( 792 ) 
-§ 14. In considering the motion in relation to the time t‚ we 
shall indicate differentiations with respect to this variable either by 
the ordinary sign or by a dot. If some quantity p may be conceived 
as a function of ¢ and likewise as one of the length of path s, we 
have the relation 
dp dp ds - ‚ds 
ENT ER? rn en 
We shall define the velocity » of the system as the complex of 
the velocities of the individual points. Its elements are av = te 
and the vector itself is 
_ ds 
y= en 4 
The direction of the velocity is that of the path, so that we may write 
OVEN ics La ata jae Vel olds ee 
and the value is 
ds 
BS 
dt 
If the value is determined by (3), the kinetic energy is easily 
found to be 
Tt mu, 
By the acceleration f of the system we understand the complex 
of the accelerations of all the material points. Thus the elements 
of f are x, , and 
— v : 
An interesting result is obtained if in this equation we use (22), 
(12) and (20). We are then led to the following decomposition of 
the acceleration into three components: 
farD4 Dae 4 Dake idee tly foD . (23) 
The first component is perpendicular to all possible displacements, 
the second has the direction of the free curvature and the third 
that of the path. 
It is easily seen that a possible motion will be quite determined, 
if we know one position, the velocity in that position and, for every 
instant, the second and the third component of the acceleration. 
Indeed, the second component determines the free curvature, and by 
this the change in the direction of the path, and the third compo- 
nent determines the change in the value of the velocity. 
« 
§ 15. Let the material points of the system be acted on by 
