¢ (3f } 
‘This being established, we may take an arbitrarily chosen part 
MN of an orthogonal trajectory of the surfaces (34), we may divide 
it into elements by means of surfaces belonging to the group, and 
infinitely near each other, and we may give to M N an infinitely 
small variation, without however changing the positions M and N. 
Then, applying what has just been said to every element of MN, 
and integrating, we find 
ofv EU ds = 0, 
showing that MN is a natural path. 
At the same time the meaning of the function & becomes apparent. 
Its value in a certain position A is the action along a trajectory, 
ending in A and beginning at the surface 
DRAA NEVE EET 
§ 26. We shall now assume that we know a function R(c) of 
the coordinates, satisfying the differential equation (33) and con- 
taining an arbitrary constant c. 
Then ie , which is itself a function of the coordinates, will have 
the same value for all positions lying on a path P, perpendicular to (36). 
To show this, we consider the consecutive surfaces whose equa- 
tions are 
Pee PaO vn EAP ea eee 
and a (oop doe) Fas ONY ies Shag) oe OGEND DEN 
Let A, be the position in the first surface where the path P 
begins, 4 any position belonging to P. We shall suppose this 
path A,A to cut the surface (38) in a position A, which is, of 
course, infinitely near Ay. We shall finally think of the path, such as 
there certainly is one, leaving the surface (38) in a perpendicular 
direction and terminating in A. It has a definite initial position 
in (38), infinitely near A,, and which we shall call 4’. 
Since 
Fil 
as may be deduced from (27), we have 
A ye Aten 
Pie Vi in an Ka ners. (39) 
Now, Ke and ae are the values of A(c) and Le 4 de) for the 
! 
position A. Consequently, the first member of (39) is the value of 
R : ave 
Cag for this position, and 
de 
