JoLY — The Associative Algebra applicable to Hyperspace. 121 

 When X or y = 1, there are only four terms in the product. For 



typifies the series, which may be illustrated on the screen by a tooth, 

 a blank, a tooth, y - 2 blanks, two teeth and 2m - y - 2 blanks. 

 "When in addition y = 1, the source of the series is + i-^ii. 

 Finally, if {12} denotes the cycle or cyclical sum, 



-of which «i4 is the source, it may be gathered from what has been 

 proved that 



P2.« = l + {12}-{13} + {14}-&c. 



+ {1234} - {1235} + {1245} - &c. 



- {1345} + &c. . . . + {123456} + &c 



The functions P2.» are sums of cyclical groups of a similar kind, 

 but of odd order in the units ; on these functions it would be tedious 

 to delay. 



59. It may be noticed that, if P is any one of these functions, and 

 ■C any cyclical sum of the units in F, PCP"^ = C. 



In particular, P%iiP~^ = %ii ; also, if m units are involved, and if 

 ,hi, ^2 . . . are the algebraic roots of h"* = 1, P'^hiiiP'^ = !SAi4, and 

 from this, various deductions may be made. 



Again, P\P-^ = i^, P'i^P-' = Vi, and P'Hy^P-"' = i^ ; 



and generally P'"p = pP'", or P'" is a scalar. 



Also, considering the linear vector function ^p = PpP~^, when m 

 is even, the symbolic equation is 



(</) + l)«-'»(<^'"- 1) = 0; 



and when m is odd, it is 



(0- l)"-'"(<^'»- l) = 0.i 



60. The following method may be used in building up, step by step, 

 the functions P : — 



Let ffia = 1 + iih, and generally qst= I + «'/'« ; 



then ?i2«i?i2"^ = h, and qi^izgn'^ = - h, 



and no other unit is changed. 



1 See Art. 38. 



