Mathematics. — “On n-uple orthogonal systems of n—1-dimensional 
manifolds in a general manifold of n dimensions.” By Prof. 
J. A. Scnourrn and D. J. Srrum. (Communicated by Prof. 
J. CARDINAAT). 
(Communicated at the meeting of October 25, 1919). 
ie 
7. Dupin’s theorem and an inversion. From theorem I we conclude 
that Durin’s theorem also holds for a general manifold: 
The Va of an n-uple orthogonal system intersect along the lines 
of curvature. 
This theorem may be inverted in the following way: 
When n—1 mutually orthogonal V,,-1-systems, determined by the 
congruences 1,...,14n—1 perpendicular to them, intersect along a con- 
gruence in, and when we can choose the arrangement of the first 
congruences in sucha way that the congruence 1, in each V,, 441i, ..,1n—1 
is a congruence of lines of curvature for the V,,_; being the inter- 
section of this Vr with the V, 11%, k=1,....,n—1, then 
1, is perpendicular to a V,,-,-system, orthogonal to the n—l given 
systems, and i,,...,14, are the congruences of the lines of curvature 
for each of the n systems. 
Proof. When the fundamental tensor °g of the V, is written: 
lil it has wee eo sal nao (7E) 
then the ideal factor a can be decomposed as follows: 
VEE EN ee Rent . (0S) 
in which a’ contains but iz, :.., in, @ but i,,..., 14-4. 
'g—a a —bb—.... is the fundamental tensor of the Vz L 
i,,...,i,—-, and the geodesic differentiation of a vector v, which is 
wholly situated in this WV, 74, is determined by. the equation: 
VE EAT AN EEND IE EEEN AE ey 
Hence for ij, we have: 
in! Viz =in1 V (ie. a)a in? Vlir.a)a Hin! V (iz. a) a’ = 
ye aaa tt WAC EN Ms 
iy © 
‚ (75) 
