210 MR GREGORY ON THE REAL NATURE OF SYMBOLICAL ALGEBRA. 



nerally said to be. The relation which does exist is due not to any identity of 

 their nature, but to the fact of their being combined by the same laws. Other 

 operations might be found which could be classed under the general head we are 

 considering. Mr Peacock and the Abb^ Buee consider the transference of pro- 

 perty to be one of these ; but as there is not much interest attached to it in a ma- 

 thematical point of view, I shall proceed to the consideration of other operations. 



II. Let us suppose the existence of operations subject to the following laws : 



(1.) fr. i^)-fn («) =f,n + „ («)' (2-) /„/„ («) = /„ „ («)• 



Where /,j,/^ are different species of the same genus of operations, which may be 

 conveniently named index-operations, as, if we define the form of / by making 

 f^ (a) = a, and suppose m and 7i to be integer numbers, we have those operations 

 which are represented in arithmetical algebra by a numerical index. For if m 

 and n be integers, and the operation a™ be used to denote that the operation a 

 has been repeated m times, then, as we know, 



a .a ^ a . (a ) z= a . 



We have now to consider whether we can find any other actual operations be- 

 sides that of repetition which shall be subject to the laws we have laid down. 

 If we suppose that m and n are fractional instead of integer, we easily deduce 



from our definition that the notation a^ is equivalent to the arithmetical opera- 

 tion of extracting the (?*'' root of the j!?*'* power of a, or generally the finding of an 



operation, which being repeated q times, will give as a result the operation a^\ 

 Thus we find, as might have been expected, a close analogy existing between the 



meanings of a'' when m is integer, and when it is fractional. Again, we might ask 

 the meaning of the operation «~'" ; and we find without difficulty, from the law 

 of combination, that a"^"* indicates the inverse operation of «"*, whatever the ope- 

 ration a may be. When, instead of supposing m to be a number integer or frac- 

 tional, we suppose it to indicate any operation whatever, I do not know of any 

 interpretation which can be given to the rotation, excepting in the case when it 

 indicates the operation of differentiation, represented by the symbol d. For we 

 know by Taylor's theorem, that 



d 



Or, a^f{x)=fix + \oga). 



In the case of negative indices, we have combined two different classes of opera- 

 tions in one manner, but we may likewise do it in another. What meaning, we 

 may ask, is to be attached to such complex operations as (+)"* or ( — )'" ? When 

 7ii is an integer number, we see at once that the operation (-1-)'" is the same as +, 



