46 E. B. Wilson, 
be expanded, each term of the expansion will contain a factor of 
the form oa, a factor of the form 8o, and no other factors which 
contains either 6 or 6. In other words, it will be possible to write 
(88) 226 (ha 6 6|6) = Ca)... (PO—c a.)|.) P)o—eeene 
where 3 is some dyadic with antecedents of one dimension and 
consequents of #w—1 dimensions. The form of this dyadic will 
depend only on the dyadic 2 and not at all on its particular mode 
of representation. That is to say, the dyadics 32), 30), ..., B("—-1) 
are invariant dyadics associated with &. 
At first it will be best to treat Z@). Let 2 be written as the sum 
Ole iy 26 BV) aS 5) sea 3 
of any number of dyads. Then 22 will be of the form 
Q=aklak+arylayt+ Briby+ 
Let the idemfactor be 7 = A44’+ uu +vv-+..., and consider the 
value of any term 
(89) (aB\aB) < (Ar Solo) = (@ BAG (a B1' 6) 
A product like «#20 or @B2'6 is called a mixed product in dis- 
tinction to the pure progressive or regressive products. The only- 
formula which will be required here is 
(90) l_a wa ve | 
18 uh vB... 
fa AS LY. iets . 
which expresses the value of the sola oer restlts from multi- 
plying any number (less than 7) of vectors into the same number 
of spaces of #—1 dimensions.} 
From the application of this formula to the case in hand there 
results 

Lie Peep. = 
(aBA6)(Aoas)= 

| ae Se B 
=(a a) (2 a) (8 6) (6 8) —(@4) (8 0) (6 a) (A B) 
— (BA) (6) ) (A a) (6 B) + (8 A) (a6) (6a) (2 B) 
=6[(a@Ai'a)B| B— (aan B)a|B—(BAX a)par(Baa Bala]. 
There is a similar expression for the other terms wu’, vv, ... of L. 
These may then be added together and simplified by the relations 
«le—ae and so forth. The result is that the contribution of 
aBlaB to Z@) is 
(aa) B\B—(ea B) a|B —(B a) Bia + (6B) ale. 
Hence finally 


' See footnote to p. 9. 
Se ee ee es a ae a ae 
