Vol. 6, 1920 MATHEMATICS: MOORE AND PHILLIPS 
157 
where Rm is an arbitrary simple space of dimension m in Mp. 
To prove this take the units ki, ^2, . .k„ so that ku ^2, . - kp lie in Mp 
and = ki2....m' I'hen 
In the summation terms containing ^^ + 1,^^ + 2, . . . .k^ maybe omitted 
since the product with Mp would be zero. The outer product of R^ 
and (2) is 
[R^{MpN,)Jo = [Rm(Mp.k,2....J{ku....m-N,)]o = 0, (3) 
all other terms vanishing because they contain repeated factors. Now 
[RmiMp.ku.... J]o = [ku....m(Mp.ku....m)]o = M p. 
Hence (3) is equivalent to 
[Mp{R^.N,)]o = 0 
which was to be proved. 
Theorem II. A necessary and sufficient condition that 
[MpN.U = 0 _ (4) 
is that either Mp and A^^ intersect in a space of dimensions > w + 1 or Mp 
contain a space of dimension p — m -\- 1 completely perpendicular to 
Nq and contain a space of dimensions g — w + 1 completely perpen- 
dicular to Mp. 
If (4) is satisfied, by (1), either R^-Nq =- 0 or the outer product of 
Mp and Rm-Nq is zero. In the first case R^ contains a perpendicular to 
Nq and consequently M p must contain a space perpendicular to A^^ 
which is cut by every R^. For this it is necessary and sufficient that 
M p contain a space of dimensions p ~ m -\- \ perpendicular to Nq. In 
the second case the intersection oi M p and Nq must contain a vector 
perpendicular to any R^ in Mp. The intersection must then be of such 
a dimension that it cuts all spaces in Mp of dimension p — m. It must 
then be of dimension m -\- 1. 
Theorem III. If 
[MpNq]^ = 0, [MpNql^ +1^0, (5) 
Mp and A^^ will intersect in a space of dimension w + 1 and so all the 
products of and Nq of index equal to or less than m will be zero ; and, if 
[MpNq], = 0, [MpNq\n-, ^ 0, (6) 
M p will contain a space of dimension p — m 1 completely perpendicular 
to Nq and so all products of index equal to or greater than n will be zero. 
In fact if relations (5) are satisfied, the vanishing of [MpNq]^ must 
be due to Mp and A^^ intersecting in a space of dimension m + 1 ; for if 
there were in a space of dimension p — m -\- 1 perpendicular to Nq, 
the product [MpNql^ + i would also vanish. Hence, in this case all the 
products of index less than m will also vanish. If (6) is satisfied it must 
be due to Af^, containing a space of dimension p — m -\- \ completely 
perpendicular to Nq, for if M p and A^^ intersect in a space of order w + 1 
