JoLY — The Associatwe Algebra applicable to Hyperspace. 119 



In order to determine the sign, note that 



jdz ■ ■ 'jm = iih ■ • • in = {-Y'^iJiH ■ • • 4^1, 

 or y,„ = ii when m is odd, and - ii when m is even. 

 The simplest form of P is 



1 + i^u = P'2, and P'^hP^'^ = h, and P^nP'f' = - %. 



The next simplest is 



P3 = 1 + 44 + 44 + 44, 



and -P3 ( ) -f*3"^ changes 4 iiito 4> 4 iiito 4? and 4 into 4- 



Instead of using the functions P'2, P\t &c., it is more symmetrical 

 to consider the functions 



P^ = hP'2, Pi = hizhP'i, &c., 



and these have the property of changing 4 i^ito 4j and 4 into 4 ; and 

 of changing to 4? 4> 4 and 4 to 4> 4; 4 and + 4> resjjectively, though 

 of course they reverse the directions of all vectors perpendicular to 

 those involved. The functions P^m are odd in the units. 



58. On reference to Art. 37, the expression for the general function 

 P-im+i is seen to be 



P-2m-\l = 1 — S*s+l^s + ^^s+l^ii+l^i'^s ~ &C., 



and the last sum consists of the sum of products of m of the derived 

 units with the corresponding original units. 



Of course, great reduction may be made on this. For instance, take 

 the series 



if t = s-^u. Assigning in this series the values 1 , 2, 3, ... m to u, 

 it is evident that 



+ 5^s*s+lV3^s+4 

 + . . . 



In this the greatest value of m is m, because s and s + 2m + I may be 

 regarded as equivalent for summation purposes, and a term such as 

 S44+i4+m+»4+m+»+i may be replaced by 244+i4+m+i-«4+m+2-«, so that assign- 

 ing any positive integer value to u, a former series is fallen back on. 



The following illustration will be of use. Imagine a cog-wheel 

 with 2ot + 1 teeth numbered consecutively, covered with a concentric 

 screen having suitable apertures. 



