( 726 ) 
T+U=E 
remains constant. 
b. In the second place we shall assume that i the varied 
motion the energy T+ U has the same constant value E as in the 
original motion. This value E having been chosen, and U being 
known for every position, the value of the velocity is given by 
i= en (E—U). 
This second assumption therefore, as well as the first, leaves no 
doubt as to the velocity with which the system is supposed to 
travel along its varied path. 
The total energy remaining constant, we have now 
0U=—dT=—mvdr, 
and (25) becomes 
Ò (vds) = d[v(0 8)s], 
or 
ÖW E—Uds) =| ymad[v(08)]. « « . . (27) 
Hence, if we integrate along a certain part of the path, supposing 
again the extreme positions to remain unchanged, 
> {YE=Uds=0 rn ste) SASS eee 
This is the principle of least action in the form that has been 
given to it by Jacopr. Indeed we may define the action along a 
path of the system as the integral that occurs in the equation (28) !). 
Its value may be calculated for any path A, 4, whatsoever. For 
the sake of brevity we shall denote it by bt 
Both the principle of HAMILTON and that of Jacosr have been 
here obtained by the consideration of the variation in the length of 
an element of a curved path that is caused by virtual displacements 
of the system. It is clear that both principles hold for every 
system, be it holonomic or not, the only condition being that the 
virtual displacements do not violate the connexions. We must however 
keep in mind that it is only in the case of holonomie systems that 
the varied motion may be said to be, as well as the original one, a 
possible motion. Hence, if we wish to compare the motion taking 
place under the action of the given forces with another motion, 
differing infinitely little from it, and such that it is not excluded 
by the connexions, the two principles will only be true for holonomic 
1) The action is usually defined as the integral, multipied by V 2m. 
