214 
STECKER. 
The geometry of Hilbert,* which lies within a closed every¬ 
where convex curve, and in which distance is defined by: 
12 = log 
foi — ^i) fa ~ ^ 2 ) 
— ^1) Oh — vj 
where and v 2 are the fixed intersections of the line with the 
closed curve, is obtained by placing: 
w (p, y — px) = 
9b 2 I 
where after differentiation b is to be replaced by y — px. 
Evidently many geometries can be obtained by proper 
choice of the function w ( p , y — px), and the author considers 
several interesting cases which we cannot enter into here. 
The function w may have singularities and a consideration 
of these leads to the important theorem: 
If w has a singularity such that over it the length remains 
finite and determinate, then the straight line still remains the 
shortest distance. 
This is followed by a consideration of the singularities of 
iv, after which this plane geometry is generalized for space. 
Since this problem is really a problem in the calculus of 
variations, let me state in conclusion an important theorem, 
given on the last page of the memoir: 
The problem of the calculus of variations remains invariant 
under every transformation , and indeed not only do Lagrange 
equations transform into Lagrange equations , but sufficient con¬ 
ditions for a minimum remain fulfilled. 
It is true that a certain determinant must not vanish, and 
that accompanying conditions in the form of differential 
equations may be adjoined. 
Ithaca, N. Y., December 16, 1902. 
*“ fiber die gerade Linie als kiirzeste Verbindung zweier Punkte.” 
Math. Ann., Bd. 34. « 
