38 



Proceedings of the Royal Irish Academy. 



operative muitiplication and division. This is advisable for the reason 

 that such a study should obviously preface more advanced work, which 

 indeed cannot easily be dealt with without it. The subject will be 

 treated in a very simple manner, because it is, in fact, such an 

 elementary one that much of it might find place immediately after 

 ordinary algebraic multiplication and division, while the rest would 

 scarcely be out of order after the multinomial theorem. It is hoped 

 moreover that some of the results may be interesting in themselves.^ 



II. Opeeative Multiplication and Division. 



1 1 . Operative Multiplication. — The terms * operative multipli- 

 cation,' 'division,' 'involution,' and * evolution,' may conveniently 

 be employed to denote the operative processes analogous to the 

 corresponding algebraic ones. Thus \jj>]x and [<^] [i/^] denote opera- 

 tive multiplication, and [<^]** denotes operative involution ; and the 

 inverse processes may be named operative division and evolution. If 

 [<^] [i/^] = Xy 4> ^ called operative factors, and x may be 



called the operative product or result ; but as (fuj/ does not necessarily 

 equal il/<f> (to drop the square brackets where the meaning is obvious), 

 we must, in the case of cj>{j/, call cl> the superior factor, and \f/ the 

 inferior one. 



The term ' operative multiplication ' is especially suitable when <^ 

 and \f/ are linear functions of jS, so that their operative product has 

 to be developed by a process akin to that of the algebraic process. 

 Thus, if 



<f> = ao + a^p + a^l^"^ .... and if/ = bo + l-^fB + hfi'^ , . . . , 

 and we have to develop their result, we must supplant every f3 in (f> 



^ So far as I can ascertain these proposals are new ; but of this I cannot be 

 sure. Professor Joly calls my attention to the fact that [c})']^ is given the name of 

 * ' the identical substitution " in the theory of Groups ; but it is equated to unity ; it 

 is not employed for the explicit rendering of operations, and does not seem to be 

 recognised as the equivalent of the unit of operation. I believe that a special 

 operative bracket has been previously suggested. The present notation was first 

 used by me, without publication, in 1886. Some of the matter given here was 

 brought to the notice of the Liverpool Mathematical Society in May, 1903 — 

 especially a paper (not included here) on the application of the method to the 

 theory of series. In this it was shown that the difference calculus is only a part 

 of a larger calculus which possesses general theorems, of which Taylor's and 

 Leibnitz's theorems are isolated examples. Expressions for these theorems were 

 given. 



