Mathematics. — “On n-tuple orthogonal systems of n—1-dimensional 
manifolds in a general manifold of n dimensions.” By Prof. J. A. 
Scuouren and D. J. Srrurk. (Communicated by Prof. J. CARDINAAL). 
(Communicated at the meeting of June 28, 1919). 
I. 
1. Notations’). A p-dimensional manifold may be denoted by V,, 
a p-dimensional euclidean’) manifold by Ze. A, may also denote 
an infinitesimal region, determined by p independent directions, in 
the vicinity of a point of V,. As original variables in a V,, we use 
the systems a and y,..., with the corresponding covariant and 
contravariant vectors 
ex = Va, ey 
9 eee aera beeen! ed (Jt) 
Sj — Vils): 
which satisfy the conditions: 
’ 
eN. = &? 3; e2.e,—e8) 7 
’ Bae | : 5 -— TT 2 
Sj - Sj — Oj H Sj.Sj = 0; 
0 when AZu 
eN eu = mai) Aro RC) 
“—=(—1l) 2 when A=wu 
0 when jk 
85. Sz’ = 
* when j=khk. 
The fundamental tensor of this V, may be written ’g: 
di, mop li Uv eers U, lS otd 1,...,n 
WE LZ Hyp eey= Z Phe ey’ = J gjrnsjsn— = pks; sy’. (3) 
dy ps do Jk Jk 
We will choose the aequiscalar V,_4 belonging to a and 17 in 
different ways according to the circumstances. 
2. Normal and V-creating fields. In a manifold V, may be given 
1) Wor the notations used in this communication see also: J. A. SCHOUTEN, 
Die direkte Analysis zur neueren Relativitätstheorie, Verh. der Kon. Akad. v. 
Wetenschappen XII, 6 (1918), here further cited as A. R. 
4 
2) We will call a manifold euclidean when its Riemann-Christoffel affinor K 
is zero. Compare A. R. p. 58. ; 
