36 



Proceedings of the Royal Irish Academy. 



It is generally assumed that [^]^ = 1 ; but, since [</)]'^^ = x. this 

 assumption implies also that 



[\~\x = \ X X, or = x^. 



That is, "we suddenly pass from a general operative relation to either 

 of the two definite algebraic relations. But what reason have we for 

 selecting these two particular algebraic relations? We have as much 

 right to maintain that 



[l~\x = \ X, or = logi^r. 

 Therefore, if [^]^ = 1, we have as much right to say that 

 \_<l>~foc = I + X, or = logi^r, 



as that it = x. 



Conversely, if i/^ be any operation such that [i//]^ = x, then, by 

 the definition oi jS, {(/ = (3. But /? is not the same as ; therefore if/ 

 is not the same as unity. And as [^]° must be included in the defini- 

 tion of ij/, then [^cfi^ is not the same as unity. In fact, as already 

 shown, [cfif = p ; that is, equals the unit of operation, not of 

 quantity, as generally supposed. 



7. We see, then, that quantities and operations are distinct entities, 

 and that a ''mixed" operation may consist of the sum of a quantity 

 and of a pure operation, just as a quaternion consists of the sum of a 

 scalar and a vector. If one term vanishes, the mixed operation 

 degenerates either into a quantity or into a pure operation, as the 

 case may be. The only operation which can be equated to quantity 

 is and this may consequently be called the zero of operation. 



8. It will be useful briefly to compare the preceding results with 

 the symbolic notation often used in connexion with the Calculus. In 

 this we have such equations as 



the expression d 



a + h ~ 



ex 



is looked upon as an operation of which the scalar element a is multi- 



plied into the subject, while the operative element — operates on it, 



ox 



each element being supposed to act after its kind. But this assumption 

 greatly limits both the power and the accuracy of the notation, because 

 both elements may have many more relations with the subject than 



