( 725 ) 
original path one that is deseribed under the action of the existing 
forces. Attending to (24), we may write in (21) 
‘ 1 1 
mov? 
Now we have (&).08)=0, because 89 is perpendicular to the 
virtual displacement. Further: 
(5-08) = mv(D.0 8) = mv (Ö 8), = mv' v() Jem (0s, 
so that (21) becomes 
O0ds=d(08), + = dv (08) — 6.08) ds, 
or, multiplied by mz», 
mvods $= 6.08) ds = md[v(0 8); }. 
The scalar product (5. 08) on the left is the work of the force 
for the virtual displacement; in the case of a conservative system 
with potential energy U, it may be denoted by — 0 U. The result 
therefore takes the form 
ded Hee see oO - + « (25) 
v 
§ 20. Thus far, we have spoken only of a varied path, but not 
of a varied motion; we have said nothing about the instants at 
which we imagine the varied positions to be reached. In this respect 
we may make different assumptions, and among these there are two, 
which lead to a simple result of the equation (25), if integrated over 
a part of the path. 
a. Let the varied positions A’ be reached at the same moments 
as the corresponding positions A in the original motion. Then 
mvdds=mvov.dt=oTdt; 
(25) becomes 
0(T—U)dt=md[v (0 8)s], 
and, if integrated along the path which the system travels over 
between the instants ¢, and ¢, in the supposition that 08 = 0 for 
t= and t= ty, 
df(t) at=0 RR aaah ee et te8) 
This is Hamitton’s principle, which is in itself sufficient for the 
determination of the motion really taking place under the action 
of given forces, and from which we may infer e.g. that in the 
course of this motion 
