( 729 ) 
paths. We may therefore call these latter the orthogonal trajectories 
of the surfaces 
Viovseaccowatir aj he EEN (1 
Let S be that one of these surfaces, to vhs a certain position 
A belongs, and let B be some position, infinitely near A, and further 
from O tan the „apmace S. In order to find an expression for the 
difference Vv jha: we consider also the surface S', likewise belong- 
ing to the group (30), and containing the position B; this surface 
will be cut in a certain position C by the path OA prolonged. If 
D is the angle between A—B aud the direction of the path in A, 
we shall have 
B A C 7 
ten Vj= Evan vo ,=V bl U. AC=V E—U. AB cos 9. (31) 
§ 23. Instead of considering the paths issuing from one and 
the same position 9, we may also begin by choosing a surface of 
positions S,, and think of all the motions in which the system 
starts from a position belonging to this surface, in a direction per- 
pendicular to it. We shall suppose that any given position A may 
be reached by one and only one of these motions, and we shall write 
pt for the action along the path leading from S, to A. 
This function has properties similar to those of the function we 
have studied in the preceding §. The paths are the orthogonal 
trajectories of the surfaces 
Ve = const., 
and the change in the action, caused by an infinitesimal variation 
of A is given by a formula of the same form as (31), 
§ 24. The values of ug and es may depend in many different 
ways on the coordinates of the variable position A, according to the 
choice of the initial position O or the surface S, from which we 
start. All these different functions have however one common property, . 
which follows immediately from what has been said, and for which 
a concise form of expression is obtained in the following way. 
If @ is a function of the coordinates, we may, for every infinitely 
small possible displacement ds, beginning in a position A, calculate 
the ratio 
dQ 
Ee TN (08) 
48* 
