Double Products and Strains in Hyperspace. 19 
duct of a zero dyadic into any dyadic is zero, and ¢ is not zero. 
Hence the necessary and sufficient condition that @ have n—/ de- 
grees of nullity is that 
(23') P= 0, Fi4.1=0. 
This condition may be applied directly to @ without any previous 
reduction. 
The geometric interpretation of the successive double powers is 
important. Suppose the dyadic is written as a sum of m terms 
with independent consequents, so that 
=eala+ B|Btylyt+. 
This dyadic converts the vectors «’, f’, 7’, ... into the vectors «, 
8, y, ..- (or their negatives).1_ The second of @ has the form 
(21) P=—aPlaBtaylayt+... 
te PVG Pear se 
Tee: 
This (secondary) dyadic converts the elements a’ f’, a’ 7", B’y',... 
of the second class into the elements @~, ay, By, .... Elements 
of the second class are the geometric counterpart of plane areas, 
namely the area of the parallelograms of which the vectors which 
correspond to the primary elements are the sides. Hence, if @ re- 
presents the transformation of vectors in space of x dimensions, 
#2, represents the transformation of two dimensional plane areas in 
that space. In like manner #; represents the transformation of 
three-dimensional volumes in the space, and so on until %, which 
gives the ratio in which #-dimensional volumes are changed. 
A considerable number of formulas for operation with double 
powers may be readily deduced: To show that 
(24) (PP), = db P, 
let ® be expressed as a sum of # dyads with independent conse- 
quents «a, 2, y, ... and let # be expressed as a sum of m dyads 
with the reciprocals of these consequents as antecedents. 
P=—ala+Psl@tyly-..., 
he Ge WG gern pa HP 
Then CA) iw ae Bey eae. 
It is merely necessary to form the expression for #2 FY, and (PP), 
to see that they are immediately identical. The same method may 
be used to show that (%#);, = #, #;, By an obvious generalisation 
it follows that the #th of a product of any number of factors is the 
product of the Aths of those factors, that is, 

17t is scarcely necessary to mention that, geometrically speaking, 
the dyadic « represents a homogeneous strain about fixed origin. 
