595 
an f,-tield by the p-vector-field of the simple p-vectors’) „v. Then 
any system of simple (n— p)-vectors, perfectly perpendicular to „v, 
determines an R,—)-field. When the field ,_,w is everywhere per- 
fectly perpendicular to a system of oo" 7 manifolds Vj,qg<>p, the 
R,-direction of JV, in each point lies in the region of ,v. In the 
case that ,_,w is not also perpendicular to a system of o”—@—! 
manifolds V,4; we call ,_,w Vynormal and ,v V‚-creating. 
When ,v=vV, ... Vp, then ‚v is V,-creating, if the p equations: 
AD ee etic. RE) 
have n—gq independent solutions. This condition is sufficient too, 
Sn the V, can always be given as intersections of n—q systems 
Ce Cea Chine, en, C, ake variable 
parameters. It is necessary that q<P, for in an R, an Rk, »4. can 
only be perfectly perpendicular to an R 
fr is also a solution of the equation: 
(5 - Y) (Vi V) fej! Vv) V fe ivy? V Vie —0, 4 7=1,.--5p, ©) 
hence also of: 
»» When « = 0. Every solution 
(CAL HRSA PW Ale Wp os ot eon o (©) 
Firstly we consider the case q — p. (In this case the system of 
differential equations (4) is a complete system.) If we write: 
n—pW —wW,:-- Wip ’ 
then we conclude from (6) that all vectors vj! Vv;—v;! Vv; lie 
in the region of »v, which also can be written as follows: 
jl,.-..p 
(v;1 Vv: — vil V vj) .we=09 (7) 
b= Ilo se opt 
If the equations (7) were not satisfied for some values of 7 and), 
we should certainly have less than 2—jp independent solutions, for 
it would be possible for these values of 7 and j to join the equations 
(6) to (4). In this way a system of 7 equations, n >r > p might be 
obtained, which would be satisfied by all solutions of (4). But this 
system certainly has more than ”— r independent solutions. Hence 
the condition (7) is necessary and sufficient. 
AS: 
ZUM (V vi) vj —(V «vy PVE eV); (Vii WI) vi a (8) 
(7) is equivalent to 
h A simple p-vectov can be written as an alternating product of p non-ideal 
vectors and can be represented by a part of a definite Fe, with definite volume, 
but of indefinite shape, together with a hyper-screw-direction, 
