Colloquium publications.

FORMS OF NON-EUCLIDEAN SPACE.

47

which is also the necessary and sufficient condition that x. may
satisfy relation (2).
Conversely any equations of the form (9) for which conditions
(10) and (8) hold represent a geodesic line, provided they are satisfied by points in T. For it is always possible to find an angle v,
such that
sin kv = X sin kl,
cos kv = X cos k1 + uo.
From the condition (3) it follows that ds2 = dv2. It can then be
verified that the functions
xi = Xyi + i
satisfy the differential equations (4).
We collect these important results in the following theorem:
Any geodesic line may be represented by the equations
x i Xy + kzi, (= O, 1, 2, 3)
where yi and zi are any two points on the line, and X and / are
parameters satisfyiing the relation
X2 + /2 + 2X/ cos kl = 1,
I being the distance between the two points yi and zi.
Conversely any equations of the above form represent a geodesic
line if they are satisfied by points of T.
From this follows immediately:
Any two linear homogeneous equations in x, represent a geodesic
line if satisfied by coordinates of points in T; and conversely any
geodesic line may be represented by two such equations.
As to the geodesic surfaces we have the theorem:
Any geodesic surface is represented by a linear homogeneous equation in xi; and conversely any such equation represents a geodesic
surface if it is satisfied by points in T.
To prove the last theorem, consider a pencil of geodesic lines
determined by two lines through Bi with the directions A' and
A' respectively. It has the equations
x. = (XA + I~A') cos ks + B. sin ks,