in the Calculus of Functions, noticed by Mr. Babbage. 445 

 since -^evidently = '^,. But \/^l ^ or - ^/ -i-^ 



(one or other of which values V^^l -}-— t must possess at 

 any time) is an e-\og of — 1, for 



^± V^i ^^^„3( j_ ^) + ^31 sin (± tt) = - 1. (22.) 



Hence, if z he positive, or, .r being real, if we choose or have 

 reason to consider the infinitesimal or zero z positive, and if 



^x = c^-^ ", I say that we shall have c 4/ — = ^l' x, what- 

 ever be c ; it being understood that corresponding powers are 



/ — 1 TT J /> v' — !• "" 



to be compared in the expressions c ^ ana c.c 



but by equation (21.) since z is now supposed positive, 



J* ^^z^z^ 



^— TTTT" ^/-*"=4,^. (24.) 



On the other hand, if z be negative, or so considered, and 

 ;|/ ^ = c^~^ % we shall no longer have 4/ x = c ^^ — , but if, 



z being negative or so considered, i> x = c^ \' T® ^^y 

 prove fn similar manner that equation (1.) will subsist what- 



ever be c. , i i i . 



We have thus therefore obtained two correlated and mutu- 

 ally complemental examples, both possessing, to a certain ex- 

 tent, the property of satisfying (1.), and both of them included 



-log l og j: 



in the generic form c'^^^-'), mentioned by Mr. Babbage 

 as derived from the process of Laplace, but m neither case 

 does equation (1.) hold good for all values of z, positive 

 or negative, nor even for all real values of x, without 

 an annexed supposition relating to those real values. Ihus 

 we sec matters so arranged, with most curious delicacy, 



that we are never at hberty to suppose v|/— = c^/.r (a 



