Double Products and Strains in Hyperspace. 17 
given for the simple case of vectors in the fifth and sixth chapters 
of the Vector Analysis. 
Given two dyadics 
P=aleat+Bl\P+ylyt+..., 
P=amlatAalatyilyit... , 
the combination 
pr P=anlamteamlepiteanilent... 
re et ig m+ A|\@AatenlByit... 
+yalyaty aly Batynivnat... 
is called the double (combinatory) product of # into #% This pro- 
duct will be denoted, as indicated, by inserting a double cross bet- 
ween the dyadics. The value of using a definitive symbol for the 
combinatory product is thus brought clearly into the foreground 
as soon as the question of these double products is taken up. 
Turning the fact that the progressive and regressive products obey 
the distributive law, it is clear that the value of p* PW does not 
depend on the particular representation of # and # which may be 
adopted. 
From the definition, the double product is obviously distributive. 
Moreover it is commutative. For the combinatory product of the 
elements is commutative exeept for a change of sign, and in the 
double product there are two changes of sign. Furthermore, the 
double product of several dyadics is associative, that is, 
(20) (2.2) 2 Sef. oO) oe wo. 
This follows from the associative property of the combinatory pro- 
duct of the elements. If a double product of more than 2 dyadics 
were formed, the laws of regressive multiplication would have to 
be brought in to determine the meaning of the product. The work 
that follows will, therefore, be restricted to the consideration of 
double products of ” or fewer dyadics. In accordance with the 
definitions given in article 4. the double product of two (primary) 
dyadics is a secondary dyadic; the double product of three dyadics 
is a dyadic of the third class; and so on. The double product of x 
dyadics is a scalar. The definition of double products may clearly 
be extended to the product of dyadics other than primary, provided 
that the class of the product does not exceed ».') 


1 It may be noted that in the Vector Analysis, p. 308, the double 
product (with a cross) of two dyadics is stated to be non-associative. 
This is because, from the point of view of the Vector Analysis, the com- 
binatory product of two vectors is not regarded as a quantity of the 
Trans. Conn. Acap., Vol. XIV. 2 SEPTEMBER, 1908. 
