( 721 ) 
The path P' that is determined by the succession of the new posi- 
tions A’ will be called the varied path and the letter 0 will serve 
to indicate the difference between quantities relating to this path and 
the corresponding ones relating to the original path. We shall use 
the sign d, if we compare the values of some quantity at the begin- 
ning and the end of an element ds of the path P. 
It is easy to obtain an expression for the variation in the length 
of an element. Starting from (1), we find 
3n 3n 
ne 
mÔds= Syma, da = Ns, me, dÒ x, = 
1 1 
3n 3n 
-x : 
== d z My u', O x, Ea ds. Dv, 2", O ty, 
1 1 
and we may simplify this by introducing the notation for the scalar 
product of two vectors, and writing (08); for the projection of 08 
on the direction of the path. The sums on the right-hand side may 
then be replaced by 
m(D.0 8) = m(d 8)s 
and m(c.0 8). 
In the last expression, in virtue of (20) 
(c.08)=(c..08)+(¢.08), 
and this is reduced to the last term, if we confine ourselves to possible 
virtual displacements 08, these being perpendicular to ¢,. Finally 
Ads AO Ds LOE ee as ox oe en) 
a result, which can be illustrated by simple geometrical examples. 
§ 13. It is to be remarked that the varied path of which we 
have spoken in the last § is not in general a possible path. This 
will however be the case, if the 7 equations (9) admit of complete 
integration, i. e. if the connexions may be expressed by # equations 
between the coordinates. 
Systems having this peculiar property are called by Hertz holonomic. 
For these, the equation (21) gives the variation which arises if one 
possible path is changed into another, infinitely near it, and likewise 
possible. 
If now the original path were one of least curvature, we should 
have cr = 0, and by integration over some part of the original path, 
in the supposition that the initial and final positions are not varied, 
2 
of as=0. 
1 
This shows that for holonomie systems the paths of least curvature 
are at the same time geodetic paths. 
