JoLY — The Associative Algebra applicable to Hyperspace. 97 



29. It is easy to show that functions of the kind described in the 

 last article are in general by no means as simple in structure as the 

 simplicity of their laws of combination might lead one to expect. 



From the equation P = §'P2^S or multiplying into g', it is seen that 

 Vq = qp. Taking conjugates Kq . JTP = - pJTq, and hence it follows 

 easily that 



Kq (P + KP) q = Kq.qp- pKq . q ; 



therefore P + KP will not vanish, unless Kq . q is commutative with 

 the vector p. 



Hence, if the operator q{ )q~'^ generates from linear vectors func- 

 tions which are the negatives of their conjugates, Kq . q must be a 

 scalar ; otherwise it could not be commutative with all vectors. The 

 converse of this is true, also. 



In this case, V^^^P = F^^P = 0, or P = ( Fd) + F"(3)) P. 



Of course, if Kq .q-x = scalar, qKq = x, also ; 



for q {Kq .q) = qx-xq = {qKq) q. 



30. Next, operating by /, the characteristic of inversion, on 



Pq = qp, and Iq . IP = piq ; 

 therefore, Iq(P - lP)q = Iq .qp - pIq . q. 



Hence, generally, if P = IP, Iq . q = scalar, and conversely. 

 In this case, F(o)P = Vf^^^P = 0, and P = ( F(o) + V^i{) P. 



31. Combining the results of the last two articles, P reduces to 

 F'(i)P, if qKq and qlq are both scalars. These restrictions on the 

 generality of q require q to be either even or odd in the units, as has 

 been proved in Art. 27. 



As an example, consider the operator depending on 



q - cos u . iiiz + sin ti . i^iii-J^} 

 It may be verified that 



q~^ = - cos ti e\4 + sin u i^iiioh' 

 For this function, qiiq^^ = q-ii = /j, and similarly I^ = qH^, while 



is = - q^iz, Ii = - q%, l5 = - q%, and Ie = - q%. 

 This very special example shows that, even when P reduces to F(i)P, 

 it cannot be assumed to be a linear vector unless some further condition 



1 This foiTa was given towards the end of the article cited. 

 E.T.A. PEOC, SEE. III. VOL. V. H 



