34 



Proceedings of the Rotjal Irish Academij. 



subject to the n^^ algebraic power. It follows therefore that \_P'\x = x, 

 that is, /? itself is an operation which produces the subject without, 

 so to speak, doing any work upon it. Now, if cf) is another operation 

 consisting of a complex of quantities combined with various powers 

 of /3, it must, according to definition, not only reproduce its subject, 

 but also perform work upon it. Thus the respective effects of /3 and <f) 

 when they operate on the same subject are comparable to the respective 

 effects of unity and any other quantity a, when these are multiplied 

 into a given element. Hence we may call ^ the univalent operation. 



Again, we have seen that y8", that is, the n*^ algehraic power of 

 has a definite potency different from that of (B. Let us now examine 

 the w''' operative power of p. Since ^ operating on a subject merely 

 reproduces it without changing it, then obviously 



that is, operative involution produces no change in p. !N'ow, if (/> is 

 an operation which produces a change in its subject, then [<^]"'^^ must 

 be different from [(j?)]''. Here again then, and <^, as regards opera- 

 tive involution, are respectively comparable to unity and any other 

 quantity a, as regards algehraic involution. 

 Thirdly, if we accept the law that 



WW" = 



then it follows that 



[_4>V [<A] = 



I^Tow, the property of as always accepted, is that it is an opera- 



tion which, so to speak, undoes the work performed by (j). Hence 

 must be an operation which performs no work on its subject, so 

 that it has the same potency as p. Hence we may write, without 

 immediate discussion, 



W~'M = W» = ^- 



This recalls the algebraic law that 



x-^x = x^ = 1. 



Comparing these several results, we shall see that /? has similar 

 properties as regards operative relations to those possessed by unity 

 as regards algebraic relations. Hence we may, perhaps, describe 

 as the unit of operation. 



