184 PROFESSOR DE MORGAN, ON THE 



mediately from the definition, in any system which gives meaning to \/— 1 

 from its commencement. 



The hardest and most delicate part of this investigation is the connexion 

 of € 6v/=I with a unit inclined at an angle 9 ; or generally to show that 

 the operation ( )^~ l changes the exponent of length into one of direction, 

 and vice versa, without the necessity of inferring this from interpretation. 

 If we assume beforehand that e^- 1 is real, under the extended definitions, 

 it would be difficult to imagine what other office ( ) v '- i could perform ; 

 but such an assumption would not be a proper one, since all the associations 

 of preceding algebra would lead us to suppose that each extension removes 

 only one class of inexplicables, and leaves, or perhaps introduces, others. I 

 cannot complete this part of the subject satisfactorily, but the following 

 considerations will show that the most simple mode of attaining, upon 

 an explanation, the technical end of the operation ( ^^ is precisely that 

 which answers to the above. 



Required an operation which repeated n times upon a function of 

 n quantities shall end by changing the sign of all. Take four quantities, 

 a, b, c, and d. Successive changes of sign made upon one after the other 

 will be really different successive operations ; but if we change the sign 

 of a given one, say the first, and at the same time make a set of periodic 

 interchanges, writing b for a, c for b, d for c, and a for d, we shall have 

 an operation which repeated four times will produce the desired effect. 

 Thus we have successively, 



<p(b, c, d, - a), cp(c, d, - a, - b), <p{d, - a - b, - c), <p{- a, - b, - c, - d). 



Thus we see in the succession (log a, 9), (-9, log a), (-log a, -9) a 

 method of passing from A to A' 1 at two similar steps, which does not 

 involve the use of y/-\. We see the same in (log or, 9), (9, - log a), 

 and ( - log a, - 9). If then we assume, as a suggestion, 



(log a, 9)^ = ( - 9, log a), (log a, 9) ' ^ = (0, - log a), 



we find, making A = (log a, 9), the following equations ; 



^v=J)^ = A~\ {A-^)'^~ l = A~\ (A^)^- 1 = A, 



i i_ _i_ 



(A^if' 1 = A'\ (A-^k)^ = A~ t , (A^M)'^- 1 = A, 



