JoLY — The Associative Algebra ajjplicahle to Hi/perspace. 103 



Consequently, the symbolic equation satisfied by <^ must be reciprocal. 

 Por the same equation is satisfied by 0'. Thus, if 



<^" - ^i</>"-i + . . . + {-Y-'M,^,<f> + (-)"3I„ = 0, 



(-)»i/;,<^« + (-)"-! j/;,_i<^"-i + . . . Jfi</, + 1 = 



is also true ; and from this it follows that M,^ = ± 1. 



N'ow, if q is an even function of the units contained in a space of 

 m dimensions (as in the recent articles), q{ )q~'^ has no effect on those 

 •of the n units perpendicular to this space, and so (<^ - l)"-'» is a factor 

 in the symbolic equation. There remains a factor of order m to be 

 considered, and as this must be reciprocal, it appears that when m is 

 odd, either <^ - 1 , or <^ + 1 must be a factor in it ; or, for some particular 

 vector (a), cospatial with the m units, either <^a = a, or (^a = - a. 



39. Generally, no other root will be equal to ± 1 ; but if ^ is a root, 

 so that (fijS =gjB = q/3q'^, it follows that g'^fS^ - ^^. If then g is not equal 

 to unity, it is necessary that /3^ should be equal to zero.^ /3 is then an 

 imaginary vector of the type (3 = a + ha', where a and a' are real per- 

 pendicular vectors of equal lengths, and where k is the imaginary of 

 algebra commutative with the units. In this case, ^• = a^- a'^ = 0, if 

 a and a' are of equal lengths and at right angles to one another. The 

 conjugate root g' must be the reciprocal of y, and as g +g' is real, it is 

 evident that g = e'"% g' = ^~'"' are proper expressions for these roots. 



It is easy to show, if jS = a + ha', that /3' = a - ha. For, suppose 

 the real vector cr is the result of operating on an arbitrary vector p, 

 by the factor of {4> - g) {(fi - g') in the symbolic equation, it follows 

 that 



(<^-/)cr = ^, and {cj, - g) a = /B' ; 



operating on these by {(j> - g), and {(fi - g'), respectively, the results 

 must vanish. But o- is a real vector, and g + g' is real, so 



(0-i(y + /))^ = i(;8 + /3') = « 



is a real vector ; also 



is a purely imaginary vector, or the product of h by real vector. 



These conjugate axes are the lines to the circular points at infinity 

 in their common plane. 



Any real unit vector coplanar with /? and /3' may be represented 



i (e'"'y8 + e-'"'fi') = acosv + a' sin v. 

 1 Compare Art. 12. 



