JoLY — The Associative Algebra appKcahle to Hyperspace. 123 



62. In the application of this method to the Theory of Siibstitii- 

 tions, it is not necessary to attend to the signs in the results of 

 operation, and considerable simplification is thereby gained. I shall 

 now give a few practical rules, in order to avoid unnecessary labour. 



If qi2 and q^i are contiguous in a product, they may be rejected; 

 for 



?12(?21 = (1 + ilh) (1 + ?3h) = 2. 



If ^12 and (/i2 are contiguous, qi^^ may be rejected ; for 



qn = 1 - 1 + 2/i«2 = 2^'l«2) 



and the operator *\4 ( ) 4"'«r^ merely changes the signs of ii and i^. 

 Thus, for substitutions, q^o, may be replaced by g'21. 



Again, as 1 + 4/3 + i^i^ + i^i^ = q-^^q^^ = q^oq^^ = q^^q^^, 



Izilvi ^^y ^6 replaced by qi^q^z, or by g'235'31. Hence, having given any 

 product Hq^t^ the first factor from the left which contains the suffix 1 , 

 say giy, may be carried towards the right till it meets a factor having 

 1 or w as a suffix. If this factor is g'^., qiuqiv i^iay be replaced by quvflw^ 

 and then q^^ may be carried on as before towards the right. If it next 

 meets g,,„, q-i^„q^,„ may be replaced by q^^q^oi, and g„i may be still carried 

 on. At last the suffix 1 occurs in no factor but on that on the 

 extreme right, q,i, suppose. If the suffix (s) occurs in any other 

 factor, it may be carried to the right until we get some factor q^t 

 immediately to the left of q^^. 



63. In this way, so far as the Theory of Substitutions is concerned, 

 the product ILq^t may be reduced to a product of the form 



^^\1st • Imt m-\ 1m-\j m-2 • • • 5'325^21 = ^l^st • ?!} 



in which none of the factors in Iliq^t are affected with any of the 

 suffixes 1, 2, ... m. The product Uiq.t may be similarly reduced, 

 so that in general (using the sign = to denote not equality, but 

 equivalence for purposes of substitution), 



n^,, = PiP2P3 . . ., 



in which the factors P are commutative ; the operators P ( ) P"^ 

 derived from them produce cyclical transposition of definite sets of 

 the letters, and the order in which the operations are performed is 

 immaterial. 



