Colloquium publications.

40

THE BOSTON COLLOQUIUM.

a region of space such that any point P of the region can be joined
to 0 by one and only one geodesic line lying in the region. We
shall call this region i. Through 0 take in T three mutually
perpendicular geodesic lines OA, OB, OC. This can be done by
taking directions (a,, a, a), (/i1, /32, 1), (71, r2 73) so as to satisfy
the relations
Ea(")la, =. 1, Ea(~,33 = 1, = a )y = 1,;. -- ik k i k i k)'Yiz
where a('~ signifies the value of ai at 0. The direction of any
geodesic line through O is then
=i alai + c2)8i + a30yi
where al, a2, a3 are independent parameters subject only to the
condition
a2+ a2 + a2 1,
which arises from
E.()i.k= i.
The direction may accordingly be named by means of (a, ta, a3).
Let P be any point on this geodesic line and let the distance
OP be denoted by r, where r is positive if measured in the direction a, and negative if measured in the opposite direction. We
may take the quantities (al, ac, Ca, r) as the coordinates of P.
Then to any set of values of the coordinates corresponds only one
point P, and to any point P correspond only the coordinates
(al, a2, a3, r) or (- a,, - a2, - ta3 - r). Between old the and
new coordinates, there exist relations of the form
Zi = Fi(aC a2 3, 'r),
where the functions F/ are continuous and possess continuous
derivatives of the first two orders since they are the solutions of
the differential equations of the geodesic lines.
By the substitution in
ds2 = aikdzidk,
the form of the line-element is obtained as