VI the Calculus ofFtttictions, noticed by Mr. Babbage. 447 



incide for a certain continuous space and divaricate in the 

 rest of their course. 



Those parts of this paper in which irifi7iitesimals have been 

 spoken of in the more popular language of mathematics, 

 may advantageously be translated into the more rigid phrase- 

 ology of //;w?V5. Various distinct continuities may terminate 

 in the same quantity as a limit, as, for example, a line may 

 be looked upon as having moved through any of the in- 

 finite number of planes of which it may be the boundary, and 

 it is easy to conceive that there are properties of such a line, 

 which (all things else remaining the same) vary with the plane 

 in which its motion is deemed to have taken place ; but it is, 

 I believe, a novelty in algebra, to present an instance of a 

 given individual function of a positive or negative quantity, 

 which varies accoi'dingly as the functional subject is regarded 

 as the limit of this or that kind of imaginary quantity. 



Professor De Morgan, in the placebeforecited, (p.335.) men- 



tions ( -j I'og (- as a form of 4/ jr not obviously disconti- 

 nuous that appears to satisfy equation (1.) independently of c. 

 We mayassumethatalwaysontheoppositesidesof that equation 



, — " ,, is intended to denote the same quantity, and that in the 

 log(-l) ^ •' 



Jogg /, 1 



" log(-l) 



expressions \\~+^l ana c \ T~ I corre- 



^°g'^ /I ^ \ 



/l_^\log(-l) / ^^\' 



KTrJ a"d c y^~^] 



X 



spending powers are to be compared. With respect to this 

 instance I shall only add that it would not be difficult to show 

 by reasoning similar to that which I have already employed in 

 this paper, that no definite case included in the indeterminate 

 logc 



expression I -— — ) can be other than a partial or 



alternative solution for 4' ^■> unless c- = 1 ; for let ~ ^ 



1 +x 



— y-V ■/ — 1 ~, then it may be proved by my exponential 

 theorems that the equation 



log c 1 log c 



\'og(-0 



X 



if, being individualized, it hold good for z of one sign posi- 



