REPORT ON CERTAIN BRANCHES OF ANALYSIS. 203 



principle of indices, and also that we ought not to say that we 

 assume a* to denote \/a, and a* to denote ^a, as is commonly 

 done *, in as much as such phrases would seem to indicate that 

 such assumptions are independent, and not subject to the same 

 common principle in all cases. 



In all cases of indices which involve or designate the inverse 

 processes of evolution, we must have regard likewise to the 

 other great principle of symbolical algebra, which authorizes 

 the existence of signs of aiFection. The square root of a may 



be either affected with the sign + or with the sign — ; for + a* 



X + a*, and — (^ x — cr, will equally have for their result 

 + a or a, by the general rule for the concurrence of similar 

 signs and the general principle of indices : in a similar manner 



a^ may be affected with the multiple sign of affection (1)^, if 



there are any symbolical values of (1)^ different from -f 1 (equi- 

 valent to the sign +), which will satisfy the requisite symbo- 

 lical conditions f . It is the possible existence of such signs of 

 affection, which is consequent upon the universality of alge- 

 braical operations, which makes it expedient to distinguish be- 

 tween the results which are not affected by such signs, and 

 the same results when affected by them. The first class of 

 results or values are such as are alone considered in arithmeti- 

 cal algebra, and we shall therefore term them arithmetical va- 

 lues, though the quantities themselves may not be arithmetical : 

 the second class may be termed algebraical values, in as much 

 as they are altogether, as far as they are different from the 

 arithmetical values, the results of the generality of the opera- 

 tions of symbolical algebra. 



This distinction may generally be most conveniently ex- 

 pressed by considering such a sign as a factor, or a symbolical 

 quantity multiplied according to the rule for that operation 

 into the arithmetical value : in this sense + 1 and — 1 may be 

 considered as factors which are equivalent to the signs + and 

 — , that is, equivalent to affecting the quantities into which 

 they are multiplied with the signs + and — , according to the 



* Wood's Algebra, Definitions. 



f That is, if there is any symbolical expression different from 1, such as 



~- , and -^ , the cubes of which are identical with 1, 



In a similar manner we may consider the existence of multiple values of l" or 

 ( — 1) , and, therefore, of multiple signs of affection corresponding to them, as 

 consequent upon the general laws of combination of symbolical algebra, and as 

 results to be determined from those laws, and whose existence, also, is de- 

 pendent upon them. 



