ON POSSIBLE SYSTEMS OF GEOMETRY. 213 
The strong (weak) monodromie axiom is said to be fulfilled if 
(J/JQ _ 
after such a circuit 9 (ds — \Zdx 2 -j- dy l > 0) returns (does 
not return) to its original value. 
Applying Lagrange’s equation, Legendre’s and Weier- 
strass’ conditions for a minimum in the calculus of variations 
to the integral I, the distance comes out in the form: 
(2) 12 — ( ----- Csin (ft — r) w (tan r, y — x tan r) dr -j- u (x % y 2 ) 
%/ COS ‘Cr 
Xi #o 
— u(x , y,) 
where ft—p, and w is an arbitrary function. 
Applying to the integral (2) the condition that the strong- 
monodromie axiom shall hold and reducing, we find: 
# 
l2f = .j*sin (ft — r) wdr. 
1 ?- IT 
A consideration of the Weierstrassian function in the neces¬ 
sary conditions that integral I be a minimum shows that if 
we take w, the simplest possible— i. e., equal to l —that we 
have: 
■& 
J*sin (ft — t) wdr = 2, 
& — jt 
and hence 12 = ds. 
This characterizes the Euclidian geometry, whence the 
important conclusion: 
From the standpoint of the calculus of variations, the Euclid¬ 
ian geometry is the simplest possible. 
If we take the function w as depending only upon the 
first argument tan r, we obtain the geometry established by 
Minkowski in his “ Geometric der Zahlen.” 
31—Bull. Phil. Soc., Wash., Vol. 14. 
