284 Prof. Guthrie on */ — 1. 



stand for x X x, and let </>(%, 2) stand for the corresponding ex- 

 ponential function, so that <j>(x, 2) = ±x <2 ._ Let the correspond- 

 ing inverse functions be denoted by */ x and <£(#, -J) respec- 

 tively ; so that whereas \/ — x* is impossible, <£( — x*, J) = +x. 

 Now, whatever be our definition of a power, it is necessary for 

 us to preserve the truth of the fundamental laws of exponents, 

 and, among the rest, that <j>(n) X <j>(m) =<j>(m + ri) ; so that the 

 following relations must be true : — 



$(#, 1)= ±x, 

 <t>{x,l)<]>(x,l) = cl>(x J 2), 



(±x) x (+#)=+#*. 



Now this last equation is undoubtedly true if we read the signs 

 crosswise, as ( + #) X (— #) or (—x) x (+#), as well as parallel, 

 as ( + #) x ( -t- #) and (— x) x (— x). Or if we read the signs 

 parallel in both cases, but invert their position in the second 

 case — since, if the signs are read parallel, ( + #) X ( + x) = +# 2 , 

 and ( + x) X (T#) = — # 2 ; so that +x is the square root of — ar 

 if the order of the ambiguous signs be reversed in the factors. 

 This affords the suggestion by means of which it appears to me 

 to be possible to dispense with the use of the impossible quan- 

 tity s/~-\. 



Let ( + )x mean that the ambiguous base which is to be re- 

 peated in a power is to be repeated with its signs in the same 

 order; but let \+x mean that base which is to be repeated with 

 its signs in alternated order, in all cases the signs having to be 

 read parallel. 



From this it will be seen that every exponential power and 

 root is ambiguous in sign, that the square root of a magnitude 

 with a positive sign is the ambiguity in which the signs of the 

 factors of the square are to be taken in the same order, whereas 

 the square root of the negative magnitude is the ambiguity in 

 which the signs are to be taken in alternating position in the 

 several factors, so that 



{±xY=x' i and (\±xy=-x\ or (±1) 2 =1 and (|±1)*=-I. 



Extending this idea, let us call any power of an ambiguity, in 

 which the alternate signs are read together, the alternating power 

 of the ambiguity. 



Then we shall have 



