210 
MATHEMATICS: L. P. EISENHART 
Proc. N. a. S. 
Suppose we have m (<w) fields of parallel vectors of components A\p), 
where i = 1, — n; p = 1, — m. If B\p) denote the components in another 
set of variables y\ we have 
B\p, = A?,, (4.1) 
If we show that there exist n independent functions 'f such that 
X,(y)^A?,)^ = 0 {i^p), (4.2) 
ox 
then in the new set of coordinates all the components of the vectors will be 
zero except those of the form B\p) . 
If we form the Poisson operator for (4.2), namely (XpXq — XqXp){y^), 
we have in consequence of (3.1) 
{X,X, - X,X,) (/) = A-J-^X,{y^) - AZ, ^-^^X.iV), 
OX ox 
where (pp (p = 1,. . .m) are the functions (p appearing in equations of the 
form (3.1). Hence (4.2) is a complete system. 
If we let p take all the values from 1 to m, there are in accordance with 
the theory of complete systems n — m independent solutions, which we 
take for y^'^^,. . . ., y". If we exclude from the system (4.2) the equa- 
tion Xr = 0 where r has a value from 1 to m, we have a complete system 
of w — 1 equations, of which n — m independent solutions are y'^^^,. . ., 
. . .y'^, and the other we take for Hence if there exist m fields of par- 
allel vectors, the coordinates can be chosen so that all the components are 
zero except those of theform A\p){p = 1,. . ., m), and consequently our 
problem reduces to the determination of geometries for which equations (3.1) 
admit solutions of this kind. 
5. In order that equations (3.1) admit solutions A\i) 4= 0, = 0 
{i 4=1), we must have 
ri, = o, ri, = ^Jog^ = (5.1) 
Ox^ \] = 1,. . .nj 
where 1^ is an arbitrary function of the x's, and then = (p/xp. Con- 
sequently if we choose the F's with one or two subscripts 1 as given by (5.1), 
and take the others as arbitrary functions of the x's, we have the most 
general geometry with one field of parallel contravariant vectors. 
In like manner any geometry with m fields of parallel contravariant vec- 
tors can be obtained by choosing 
rW=o r|, = ^iog^, C'K^ ^^-^^ 
where xpp is an arbitrary function of x^, x^'^^,. . ., x^. 
