MR GREGORY ON THE REAL NATURE OF SYMBOLICAL ALGEBRA. 211 



but ( — )'" becomes alternately the same as + and as — , according as m is odd or 

 even, whether they be the symbols of arithmetical or geometrical operations. So 

 far there is no difficulty. But if it be fractional, what does (+)"* or ( — )"" signify? 



In arithmetic, the first may be sometimes interpreted, as because (+)"* = + when 



1 \_ 



m is integer, (+)»* also = +, and as ( — )2'» = +, also (+)2'» = — : But the other 



symbol ( — )"* has, when m is a fraction with an even denominator, absolutely no 

 meaning in arithmetic, or at least we do not know at present of any arithmetical 

 operation which is subject to the same laws of combination as it is. On the other 

 hand, geometry readily furnishes us with operations which may be represented 



i_ 1 



by (+)"* and ( — )*", and which are analogous to the operations represented by + 



and — . The one is the turning of a line through an angle equal to -th of four 



right angles, the other is the turning of a line through an angle equal to -th of 



two right angles. Here we see that the geometrical family of operations admits 

 of a more extended application than the arithmetical, exemplifying a general re- 

 mark we had previously occasion to make. Whether when the index is any 

 other operation, we can attach any meaning to the expression, has not yet been 

 determined. For instance, we cannot tell what is the interpretation of such ex- 



d d 



J — ^ ^'^S 



pressions as (+)'*'' or ( — y'', or {+) . 



III. I now proceed to a very general class of operations, subject to the fol- 

 lowing laws : 



(1.) /(a)+/(6)=/(a + 6). 

 (2.) ^/(«) =/^(a). 



This class includes several of the most important operations which are considered 

 in mathematics ; such as the numerical operation usually represented by a, b, kc, 

 indicating that any other operation to which these symbols are prefixed is taken a 

 times, b times, &c. ; or as the operation of differentiation indicated by the letter d, 

 and the operation of taking the difference indicated by a. We therefore see what 

 an important part this class of functions plays in analysis, since it can be at once 

 divided into three families which are of such extensive use. This renders it ad- 

 visable to comprehend these functions under a common name. Accordingly, 

 Servois, in a paper which does not seem to have received the attention it de- 

 serves, has called them, in respect of the first law of combination, distributive 

 functions, and in respect of the second law, commutative functions. As these 

 names express sufficiently the nature of the functions we are considering, I shall 

 -use them when I wish to speak of the general class of operations I have defined. 

 It is not necessary to enter at large here, into the demonstration that the 

 symbols of differentiation and difference are subject to the same laws of combina- 



