212 
STECKER. 
tion of Non-Euclidian Quadric Surfaces ” by Mr. Coolledge, 
of Harvard, read before the American Mathematical Society 
at New York, and which is to appear in the transactions of 
of the society; also to* a paper of my own, in the current 
number of the American Journal of Mathematics, under the 
title “Non-Euclidian properties of plane cubics and of their 
^first and second polars,” in continuation of some former re¬ 
search work.f 
(d) In conclusion let me call attention to the important 
results of G. Hamel, a pupil of Hilbert’s, under the title 
“ Uber die Geometrien in denen die Graden die Ktirzesten 
Sind.” J 
This question was raised by Hilbert in a letter to Klein, 
afterwards published in volume 34 of the Mathematical An¬ 
nals in the year 1889.. Certain types of such geometries were 
considered by Minkowski in his “Geometrie der Zahlen” in 
1896. 
Hamel considered the subject as a converse problem in 
the calculus of variations. 
His first result is that if a certain direction is given to a seg¬ 
ment, its length can always he represented as an integral taken 
along that segment: 
a ? 2 
I- 12 = fg (y, X, y') dx, jy = -jyj 
X 1 
where g ( y , x, y r ) is in general, within the domain considered, 
finite and continuous, and possesses first and second deriva¬ 
tives with regard to all three arguments. 
The matter does not require that g taken from to x 2 along 
a segment is the same as taken from x 2 to x x along the same 
segment, so that starting at a certain point, passing to + oo, 
and returning through — oo along the curve, g may not re¬ 
turn to its original value. 
*Am. Journal of Math., vol. xxiv, p. 399. 
t Am. Journal of Math., vol. xxii, p. 31. 
t Gottingen Thesis. 
