( 728 ) 
This follows from what has been said in § 5, if we consider that 
AC is the resultant of the displacements A— B and BoC, 
§ 22. Henceforth we shall treat only of natural motions, taking 
place with a fixed value Z of the total energy, which we choose once 
for all. We shall suppose that, if O and A are any two positions, 
there is une and only one such a natural motion which leads from 
O to A, The action along the path of this motion, 
7 A 
ee =f VED, 
will have a definite value, depending on the coordinates of O and A, 
and we shall examine the variations of this action, if we change 
the final position A, the initial one being fixed. 
In the first place it is clear that, if we move A along a path 
A 
issuing from 9, V will be the greater, the farther A recedes from 0. 
A 
Indeed, V, presents a certain analogy with the length of the path, 
the difference being that, in calculating the action, we must mul- 
tiply each element ds by the factor Y~ #—U, which changes with 
the position. 
The increment of the action, corresponding to an element of the 
path, is obviously 
VY E—Uds. 
In the second place we compare two paths, both issuing from 0, 
but in directions that differ infinitely little from each other. We 
shall proceed along these so far, say till we have reached the posi- 
tions A and A’, that the action is equal in the two cases, 1. e. 
0 0 ee 
Now, the motion O— A’ may be conceived as the result of an 
infinitely small variation of the motion OA; we may therefore 
apply the equation we deduce from (27), if we integrate from O to 
A, On account of (29), we get 0 on the left-hand side, hence, the 
projection (0 8),, which vanishes for the position 0, must likewise 
be 0 for the position A, and the infinitely small displacement A — 4! 
is found to be perpendicular to the path OA. 
We may next fix our attention on all paths that issue from a 
definite position O. In each of these we choose a position A at such 
a distance from O, that the action Vo between O and these positions 
has the same value for all of them. The positions A will belong to 
a certain surface and this will be cut under right angles by all the 
