JoLY — The Associative Algebra appllcahU to Hyperspace. 93 

 Suppose, in the first place, that (^o = ^o is not zero, and let 



as before, and a^q^ = (iiix^ = Hqiq^ + q^qx)- 



The first condition of commutation is satisfied identically, and the 

 second becomes 



«i2^'i4 ■ hfi - C'niil^ - i\H = Oj 01" <*5i2 {i^f^ + (Sio) = 0. 



Provided «i2 is not zero, this requires (3 = a2shhi Siud therefore 



$'3 = «0 + «i?i + «i2h«2 + «23 his + «o"^'^23«'l4«'3 ; 



and this is a function of but three units, and satisfies q2^ - q^ = scalar. 

 If «i2 is zero, the conditions of commutation are satisfied identically • 

 but qz - qi = (^'{1 + a^^a-^) requires (S^ to be a scalar, and /3 = «2344 is 

 a necessary form (see the last article). The cubic is now 



If «i = 0, so that qx = 0, «o or q^ must likewise be zero, if the function 

 is to remain a cnbic. 



The conditions now to be satisfied are 



q^qz - q^q-i = O, and q-^ - qi = scalar. 

 These conditions are satisfied for the cubic in five units, 



q = a\ii\i% + azibiziJi, 

 and doubtless for other forms also. 



If q^ = 0, either q^ = 0, or qi = 0, which is the case just considered. 

 If 5'2 = 0, (7i$'3-Mi = shows that, if qi = aih, $'3 = «i/3 = «i23«"i Vs, 

 because /3^ must be a scalar. 



25. Eeturning to the symbol /of Art. 17, for which, if 



S = ^(0) + S'(l) + Si2) + ?(3), Iq = !?(0) + Sih - ?12) - ?(3), 



the conditions of commutation of q and Iq are contained in the equation 



(^(0) + sw) (2'(2) + Si3)) = (!Z(2) + !Z(3)) (2'(o) + ^(1)) ; 



or, on separating even and odd parts, 



2'(o)2'(2) - 2'(2)2'(o) = 2'(3)2'(i) - 2'(i)2'(3)> 



and ?(o)2'(3) - 2'(3)2'(o) = ^(2)2'(i) - ?(i)i2'(2j ; 



or again, 



^(0) (2'(o)2'(2) - ?(3)$'(i)) = 0, and Fd) (!?(o)^(3) - 2'(2^!Z(i)) = 0- 

 (Cf. Art. 20.) 



