Double Products and Strains in Hyperspace. 13 
If, however, elements were inserted between the dyadics in the 
product, the associative property would be lost. 
If @ and W are two dyadics which have respectively the nullities 
n—k and n—-/, so that 
@=ylat+AlAat...tuala (k terms) 
and P= ay|co + BolBot...+ AlAs (/ terms), 
the product #¥ or the product #@ cannot have a nullity less than 
the greater of the two nullities z— and n—J/, nor a nullity greater 
than the sum, 2 7--k—/, of the nullities. To show that the nullity 
of ¥ is at least m—/, it is merely necessary to inspect the product 
PP = (P aty)| @z + (P Bo) | B2 + ... + (PAs) | do. 
To show that the nullity is at least as great as m—k, consider 
P= a4|(HP+AlAHM+..-+ala DP. 
To see that the nullity of # ¥ is not greater than 22—k—/, con- 
sider the antecedents @a,, ®~,, ..., @A, in the first expression of 
the product. As the nullity of # is only m—A, not more than ~—k 
dimensions of space are annihilated and hence in the most un- 
favorable case at least /-u+k of these antecedents must be line- 
arly independent. Hence the nullity of @¥ is not greater than 
n—Ilt-n—k, and the proposition is proved. It is, of course, obvious 
that the nullity of the product could not be greater than 7. With 
this understanding the generalisation to a product of any number 
of factors is immediate. The theorem is due to Sylvester. 
If either # or ¥ has any degree of nullity, the product cannot 
equal the idemfactor /, which has no degree of nullity. Dyadics 
which have no degree of nullity will be called complete. If # and 
Y are two complete dyadics which satisfy the relation PY — J, 
they will be called reciprocal dyadics— 
(43) PP —=[— VO, P=, b= VP, 
It may be shown that in this case the product of # and is com- 
mutative as indicated and that the reciprocal of either is uniquely 
determined. 
The reciprocal of a product may be shown to be the product of 
the reciprocals taken in inverse order, that is, 
(14) (PPQ)1 = 2-1 w gr, 
The reciprocal may be written down immediately. For if 
(15) d= alat plg+...+ v|D, (—1)"-1@-1=¢'|a’' + # et ieee by |e. 

1 The proofs of these statements are so simple and so like those given 
for the simpler case in the Vector Analysis, chapter 5, that there is no 
need of giving them here. The same is true of a large number of pro- 
positions which follow. See also Whitehead’s Universal Algebra. 
