Colloquium publications.

42

THE BOSTON COLLOQUIUM.

and we proceed next to replace dO by its value in terms of ai.
For that, we call 80 the angle between two neighboring geodesic
lines OP and OQ, with directions a and a + 8a, where
1 2 3
a + a2 + a2 1,
(a, + 8a,)2 + (a2 + 8a2)2 + (a3 + 8a,)2 = 1.
Then
cos 80 = al(a1 + 8al) + a2(a2 + a^) + ca3(a + &a3)
- 1 -+ eSal + a28a2 + a 38a3
- - 1(8a2 + 8a2 + 8a ).
so that
sin 2 0= 4(Sa2 + 8 8a2 + )
From this follows in the differential notation
d02 = Cda2 4- ca2 + da,
So that the line-element of the suface is
(ds2 = G(da2 + lda2 + da2) + (,1.2
This is in particular the element of the length of the curve C,
since C is on the surface. But C is any curve in space and hence
the above expression is the line-element of the space.
We seek now to determine G. For that purpose consider
G = Al, = Eaik __
where (see p. 40)
Xz = F v(al,, 3, r)- = z + (alai + a2/i + a3yi)i +
Hence
G = r'2Eaik aa +..
and consequently
(1/ C~)=O0 = o - all,=0
Thus far the discussion is applicable to any space which satisfies
the first two hypotheses. We examine now the effect of introducing the third hypotheses. A geodesic surface formed by a