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MATHEMATICS: MOORE AND PHILLIPS Proc. N. A. S. 
as above. We shall now show that the product of two such quantities is 
invariant, i. e., that it is independent of the system of units in terms of 
which it is expressed. We do this by showing that [MN]m can be ex- 
pressed in terms of invariant operations on M, and the identical trans- 
formation^ (dyadic) I. 
If ki, k2, . . . . kn are the unit one- vectors, the 1, 1 idemf actor is 
II = kiki -\- ^2^2 + • • • • + knkn- 
Using the dot to express inner multiplication we then have 
kl2.Il = kik2 — hki = —Il.ki2, 
knsJl = kiski — ^13^2 + ^12^3, 
Il'km = kik23 — ^2^13 + ^3^12, 
and similar expressions for products with units of higher order. Using 
these values we find 
[kl2ku]i = [{ki2Jl).{Il.kiz)]o, 
where the subscript 0 indicates that the antecedent and consequent of 
each dyad in the brackets are multipled according to the (outer) multi- 
plication of index zero and the sum of terms then taken. Similarly we find 
[kmkuf,]l = [(Ali'3./l).(/l.^l45)]o, 
and generally 
[7ri7r2]i = [(7ri.Ii).(Ji.7r2)]o 
where tti and 7r2 are any product of units. vSince the operations are all 
distributive we have then 
[MN]i = [(M./i).(/i.A^)ol. 
Since the dot multiplication, the outer (zero) multiplication and Ii are 
invariant it follows that [MN]i is invariant. 
For the product of index 2, we use the 2, 2 idemf actor 1 2 = ^kijkij with i<j. 
We then find 
[^123^124]2 = [(^123./2).(/2.^124)]o 
and generally 
[7ri7r2]2 = [(7ri./2).(/2.7r2)]o, 
or since the operations are distributive 
[MN]2 = [(M.l2).(/2.N)]o 
which shows the in variance as before. 
In general the product of index m is 
[MiV]„ = [(M.IJ.(I„.iV)]„ 
where 
Im — 2^12 VI ku m- 
Let be a simple space of dimension p and A^^ a simple space of di- 
mension q. 
Theorem I. In order that [MpNq]^ = 0 it is necessary and sufficient 
that 
[M,(i?,„.iV,)]o = 0. (1) 
