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MA THEM A TICS: MOORE AND PHILLIPS Proc. N. A. S. 
the product [MpNq]„-. i will also be zero, hence, all the products of index 
less than n will vanish. 
It is clear from the theorem just proved that all the products [MpNq]r 
for which m<r<n will be different from zero and all the products for 
which r<m or r>n will be zero. 
Theorem IV. If equations (5) are satisfied, [MpNq]^^ + i 
is the outer product of the parts of MpNq completely perpendicular to the 
intersection. 
In this case Mp and Nq intersect in a space of m + 1 dimensions. Let 
ki, ^2, . . . + 1 be unit vectors in that space. Then 
Mp = kik2. . .K + \P 
and 
N = kih ■•■km + \Q 
where P is the part of Mp perpendicular to kiki . . .k^ ^ i and Q is the part 
of Nq completely perpendicular to kiki. . .k^ + i- Hence 
[MpNq]^ + , = [PQ]o 
which proves the theorem. 
Theorem V. If (6) is satisfied [MpNq]n~ i is the outer product of the 
part of Mp perpendicular Nq and the part of Nq perpendicular to Mp. 
Let lu I2, . . . .lp — „ + I he the units in Mp perpendicular to Nq and ;i, 
j2, . . . .jq—n + i the units in Nq perpendicular to Mp. These units are 
then perpendicular to each other and 
Mp = Pkh. . . .Ip-n+l 
Nq = 2/1/2. . . .]\-n + l 
where P (of dimension n — 1) is the part oi Mp perpendicular to I1I2. . . 
„ + 1 and Q is the part of Nq perpendicular to /1/2. . . ./g_„ + 1. Since 
the units kk - . . .Ip — n + ly jih- • • - jq— n + \ are all distinct 
[MpNq]n-i = [PQ]n~\[kk. + 1/1/2 /g-n+l]- 
Since [PQ]„ + i is a number this proves the theorem. 
^ Gibbs, Wilson, Vector Analysis. 
2 Proc. Amer. Acad. Arts Set., 46, 1910 (165-181). 
3 The products here discussed occur as partial products in the multiplication of 
generalized quaternions. From this point of view they have been treated by Clifford 
(Amer. J. Math., l), Joly (Proc. Royal Irish Acad., 5, (3)) McAuley (Proc. R. Soc. 
Edinburgh, 28), Shaw (Bull. Int. Assoc. Adv. Quaternions, 1913, (24-27)), and others. 
In victor algebras of the Grassmann type, for example in the usual victor theory of 
relativity, they have, however, such a different appearance that we have presented 
them here from that point of view. 
* In case the index is equal to the order of one of the factors the sign is taken as in the 
Lewis inner product. 
5 We used this as the definition of the product of index 1 (star product) in a paper 
"Rotations in Space of Even Dimensions" to appear in Proc. Amer. Acad. Arts Sci. 
