Voh. 7, 1921 
MATHEMATICS: L. P. EISENHART 
329 
is of one of two types: 
(I) ds 2 = d%i 2 + adx 2 2 + 2pdx 2 dx s + T^3 2 , 
<*, (8, 7 being functions of #i alone, the operators of a the group being 
Xi =b/bx 2 , X 2 = d/dxs; 
(II) ds 2 = dxi 2 + adx 2 2 + 2(0 - a* 2 ) + (ax 2 2 - 2(3x 2 + t)^ 3 2 , 
a, jS, 7 being functions of #i alone, the operators of the group being 
Xi = d/d* 3 , X 2 = ^ X3 c)/c)x 2 . 
2. By definition 
0 0 
so*} -{i }{?})]• 
(2) 
0 
where the Christoffel quantities are calculated with respect to the linear 
element of the space-time continuum. When this linear element is taken 
in the form V 2 dx Q 2 — ds 2 as in § 1, the equations = 0 reduce to the 
six equations 2 
bxi 
and 
2« tt *»-b, (4) 
where a* fe is the cofactor of a# in the determinant of the quantities di- 
vided by this determinant ; the Christoffel symbols are calculated with re- 
spect to (1), and 
3 3 3 
B ik = 2^ { ih > hk } = 2^ 2 Z °" (* (5) 
l i i 
By assuming that 5 is referred to a triply-orthogonal system, so that 
3 
the linear element may be written ds 2 = i aidx? (which is possible in 
l 
any 3 -space 3 ), and making use of the relations 
{il,hk) = - (li, hk) = - {il, kh) = (U, kh), 
we find that when Bik — 0, then all the functions (il, hk) are equal to 
zero. This being the condition that the space S is euclidean, we have 
the theorem : 
A necessary and sufficient condition that a 3-space be euclidean is that the 
functions Bik = 0. 
