336 Mr. Graves's Explanatioii of a remarkable Paradox 

 mean A- and — as well as 1. But I think that, in the first 



V V 



place, such a shifting is not a legitimate proceeding, for the 

 different values of a generic function of .v are themselves dif- 

 ferent individual functions of .r, while the equations in question 

 suppose a comparison of the same functions, and are couched 

 in an algebra applying to individual values ; and, in the se- 

 cond place, even such an explanation would not succeed in 

 the case before us in wholly removing the real difficulty, for, 

 even by the utmost shifting, we cannot suppose c an entirely 

 arbitrai-y constant, unless for any given .r, i> x itself be wholly 



arbitrary, which c ^^sC-^) assuredly is not. The true so- 

 lution resembles while it differs from that thrown out by 

 Mr. Babbage, unless (as may possibly be the case) the vague 

 terms he uses are intended to adumbrate the very sohition 

 which I contend to be con-ect. The paradox is occasioned 

 not because \J/ a- has different values for one value of x, 

 but because ^ has different forms for different values of .r. 

 Is this so, and, if so, how does it obviate the difficulty ? These 

 are the inquiries which it is the design of this paper to answer. 

 At the very threshold of our researches a question that 

 must present itself is, What do we mean when we put ;{/ x 



- log'^ X 1 



_ - logC-O, and assert that ^x= c^ — ? Do we mean 



- log' X -J°g' I 



to assert that the expressions c °^^~^'' and c.c'°s(-'^ are 

 equivalent generic forms, (like a^ h^ and (a h)^) as having each 

 the same infinite number of values for any given x, when we 

 let in all the ackuffwledged indefiniteness of the notation log 6 

 and a^? As I presume that such an assertion of general 

 equivalence is not intended, and as it is not necessary for my 

 future argument to deny it now, I shall not stop to prove its 

 incorrectness, but this, at least, I think we must be supposed 

 to mean, viz. that, for any given x within certain limits, the 



— log' X 



form c ^°S (—1) includes some values or some fixed single 

 value which respectively are or is equal to other severally 

 corresponding values or another corresponding individual value 



- log--j 



included in the form c.c iog(-iT. I proceed, therefore, to 

 point out a case oi^x included in the form c ~ logC-i)^ which 



