( 730 ) 
The value of this will of course depend on the direction of d8, and 
the position A being chosen, there will be one definite direction 
(perpendicular to the surface Q=const.), for which the ratio takes 
the largest positive value. Now, if we denote this maximum value 
of (32) by DQ, the property in question of the functions V may be 
expressed by 
DV, =V £—U and DV, = VED 
These formulae may be written in the form of a partial differential 
equation which is satisfied by V, and V 
§ 25. Let R be any solution of this differential equation, i.e. a 
function of the coordinates, such that 
DRS VEU oe. ek See 
then the orthogonal trajectories of the surfaces 
Venen toten ven el ne en 6S: 
are natural paths of the system. 
The proof of this is as follows. Imagine the infinitely small dis- 
placements d8, lying between the two consecutive surfaces 
R=C and R=C4d, 
that is to say, the displacements whose initial position belongs if 
the first and whose final position belongs to the second surface, and 
subject them to the further condition that they are to be bonne 
to the first surface. Then we have by (53) 
sn EA VY E—Uds = dc, 
so that the action is the same for all these elements, whichever be 
the position in the surface from which they start. 
On the contrary, if 78’ is another element of path between the 
two surfaces, not perpendicular to them, but making an angle J 
with one of the first-named displacements d 8 in the immediate vici- 
nity, we shall have 
and for the action along d38' 
dC 
cos S 
It appears from this that 
IVER 0) i) a eee 
if we pass from an element 8, ele 6 R= C, to an 
element d@8', lying between the same two surfaces, and of ares a 
direction that # is infinitely small. 
