D O C U M E N T 4 9 F E B R U A R Y 1 9 2 1 4 5

assumes the factor only if the ’s are substituted by . Ac-

cording to Herglotz,

But here:

This situation immediately suggests considering the quadratic formula

,

which also depends only on the ratios of the ’s, and setting the variation prob-

lem

with firm boundaries.

In the conventional scaling one finds the differential equations

(A.)

[7]

For a of constant curvature and also I constant and other than zero, when

the curvature is other than zero, the geodesic lines therefore remain the same here

too. If the curvature is zero, however, then I = 0 and the transition from to

is impossible.

For spaces of constant curvature, vanishes identically as well.

One easily verifies with the formula

that the differential equations (A.) remain unchanged if the ’s are replaced by

.

Equations (A.) thus, in any event, pose an interesting generalization of geodesic

lines, in which they gradually change to constant curvature and in which they hard-

ly deviate in a region where the differential quotients of I are small. I do not dare

to decide whether physical meaning should be attached to the additional terms

linked to I.

I hope this message reaches you in the best of health. With many thanks for the

rich stimulus in Vienna, yours very sincerely,

Wirtinger.

H K

1

n 2 –

----------- - g K g K – g K g K – + + =

1

n 1 – n 2 –

- g g g g – K –---------------------------------

1 a + g 1 a + g

g K K , = g K K. =

H g 0. =

d 2 ds2I1 /2 =

g

d 0 =

2

2

d

d xr ik

r

d

dxidxk

d

1

2

-- -

d

dxr

dlgI

d

--------- -

1

4

--I–1 - /2

logI

xl

------------glr

l

– +

ik

– =

ds2

ds2 d 2

H

ik

r

1

2

-- -

xi

a

kr

xk

a

ir

gik

xh

a

ghr – + =

g

1 a + g