32 



Proceedings of the Royal Irish Academy. 



the subject of the operation. "WTiat we have to do is to find an 

 expression for the operation apart from its subject. Tor this purpose 

 we must obviously retain the whole form of the original complex 

 (without which its structure cannot be rendered), and at the same 

 time eliminate the subject. IS'ow, if we simply erase the subject, we 

 shall lose count of its relations with the other members of the group. 

 We must therefore employ a symbol to denote its position within the 

 complex. Let us call this position the hase of the complex (upon 

 which it has, so to speak, been constructed), and denote it always by 

 the symbol p. Then, putting (S in the place of the subject-symbol, 

 we have a new expression which exactly represents the operation 

 apart from its subject. The representation is exact, because all the 

 facts contained in the original expression — the values of the subsidiary 

 elements, and their relations with each other and with the subject — 

 are retained. At the same time, because it contains an element 

 which has no quantitative value, the new expression has none, and 

 cannot therefore be equated to any quantity. It is, as it were, a 

 shadow-function, possessing the form without the material of the 

 original. It expresses a definite algebraic action^ and may be called 

 shortly an explicit operation, or, perhaps, a verl-function. 



For example, the action performed in the construction of the 

 quantitative or scalar function a ^Ix \% a ^ hp — those performed in 



e'^ e^ 



the construction of — and x eos-^x are —-p: — — and /S cos'^/S, 



x{x-l) f3{f3-l) 



respectively. If convenient, we may denote any of these by a single 

 operative symbol : thus we may write cj> = a + hj3, ^ = If 

 the operation is to be performed on two or more subjects, we may 

 distinguish the several bases by accents or subscripts, as in 



3. It is obvious from the definition that verb-functions are capable 

 of any algebraic or other relations of which scalar functions are 

 capable. Thus /3 cos-^/3 is the algebraic product of the operations 

 /3 and cos~^/?, and 



L_ 



Moreover, explicit operations may combine algebraically with implicit 

 operative symbols. Eor example, when a -i- bjS + cf) operates on x, 

 the result is 



a + hx + (f> {x). 



