in the Calculus of Functions^ tioticed by Mr. Babbage. 337 



satisfies the equation '^ix = c>\) — ; but I hope to show that " it 



satisfies that equation only for such values of x as to exclude 

 one of the assumptions we must make if we would prove c^= ]," 

 or that, " if it satisfies equation (1.), x must be classed with 



one set of values, and — with another different set : and 



X 



further, that if any of the latter set be substituted for x in the 



— 10ff2 X 



particular case of vJ/cT included in c ^°& (-^^ which before sa- 

 tisfied (1.), that equation will no longer hold good, so that 



(2.), which is obtained by substituting — for x in (].), can- 

 not subsist simultaneously with (1.) for the assigned form of 

 \I/x." The instance I shall select will be given as being the 

 simplest illustration I can present of the application of this 

 principle, and as being, morever, independent of any hitherto 

 disputed peculiarities in my results concerning logarithms; — • 

 not that 1 at all recede from those results, but that I wish to 

 approve my present results to those who are not yet convinced 

 of the former. The same principle may be proved to apply if 

 we select any other individual case of \(/a; included in the form 



log" I 



c ~ l°g(-0, which when ^ is within certain limits, satisfies 

 equation (1.), when c^ is not = 1. 



To prevent misconception, it may be as well here to re- 

 mark, that I admit no difference of meaning in result^ what- 

 ever difference of intermediate operation may be denoted, be- 



— log* g log"* log* X 



tween c l°g (->), c~ >og(-i) and c -»«'S(-i), and that as a ge- 

 nericform, I considers ^°«(-i) to be precisely equivalent to 



— log' I 



^log(-i)^ for if ^ be a logarithm of — 1, so also is —k. 

 Hence both expressions contain exactly the same values, it 

 being remembered that any value common to both corre- 

 sponds to different individualizations of log ( — 1) in each. 

 The proposed functional solution for \|; being included in 



log' X 



the form c^°s(.-i) leads us to consider algebraical symbols as 

 capable of possessing imaginary values, since it involves 

 log (—1), an expression which has no real value. In the pre- 

 sent case we shall see that the discussion of the proposed func- 

 tion of X is cleared from the obscurity that surrounds it when 

 X is real, by our knowledge of its properties when x is infini- 

 T/iird Series. Vol.9. No. 55. Nov. 183G. z K 



