( 720 ) 
z 
my, a", ZE : t Uy P; y ° e e e . e . (19) 
1 
and the formulae (18) will serve to determine the quantities P,. 
A possible path which, in each of the positions belonging to it, is 
less curved than any other possible path of the same direction may 
be called a path of least curvature. In every position through which 
it passes it has the property expressed by (19) or, as we may also 
say, its curvature is perpendicular to all possible displacements. 
A path of least curvature is determined by one position, and the 
direction in that position. 
§ 11. We shall next consider a possible path P and the path 
P, of least curvature, having in common with P one position A and 
the direction in that position. Let, in the position A, ¢ be the 
curvature of Po, yo) the elements of this curvature, € and a", the 
corresponding quantittes for P, and let us fix our attention on the 
relative curvature of P, with respect to the least curved path Po. 
We shall denote this relative curvature by cy, and cali it the free 
curvature of the possible path P. It may be shown to have the 
direction of a possible displacement. 
Indeed, we have by definition 
Cy = C — Co Peter Sag (20) 
so that the elements of cr are #’,—a"yo). Now, if we write down 
two times the equations (18), first for P, and then for P, we find 
) 
by subtraction 
3n 
Set [a",—a"o)| = 0, 
1 
which proves the proposition. We may add that ¢, is perpendicular 
to cy, being perpendicular to all possible displacements, and that 
therefore by (8) 
iN de er" 
This confirms the inequality ce, < c. 
It is easily seen that a possible path is wholly determined if one 
knows one position belonging to it, the direction in that position and 
the free curvature in all positions. 
§ 12. Let P be a possible path. From every position A lying in 
it we make the system pass to a varied position A', by giving to it 
an infinitely small displacement 08, for whose elements we write dy, 
these elements being supposed to be continuous functions of the length 
sof the path, reckoned along P. 
