in ike Calculus of FuncHons, noticed by Mr. Babbage. 339 



When once we introduce into our calculations the considera- 

 tion of imaginary values, we have to treat x o^ y ■\- \/ i ~ 



as essentially a variable in both its constituents, and therefore 

 z can never fairly be supposed in the condition of an absolute 



central stationary zero. Let I a/ j/^ + z^ denote the arithme- 



-1 y 



tical Neperian logarithm of v^y + s*. Let cos^ , ~ r - 3 -3- 



denote the circular arc not less than + and not greater than 

 !r, which, when radius = 1, is equal to , ■ -. It is ob- 



vious that such an arc is always assignable, since 



can never be greater than 1, nor less than —1. 

 I come now to the following proposition, before referred to, viz. 



g _i y 



^^l Vf- + z'~ + ^-1:7^ cos^^ Vy+P 'saNeperian 



logarithm or e-log of :i'." This proposition is proved in my 

 papers on exponential functions cited in the 8th volume of this 

 Magazine, p. 281, (Number for April 1836) ; but as the heads 

 of the proof are very simple and at the same time illustrate 

 my subsequent reasoning in the present case, I shall briefly 

 recapitulate them here. 



Let the notation e^ denote that particular value of the am- 

 biguous expression e\ which is represented by the sum of the 



series , . A" 9^ 6» 



^+^+P2 +1723- + LsTTn- (^-^ 



where 9 is not limited to real values. It has not been unusual 

 to treat of e^ when 9 is imaginary, and when this has been 

 done, the meaning of e^ must have been tacitly defined (though 

 probably without regard to the possibility of e having many 

 values for any given 9) by reference to the preceding series, 

 which is convergent for all values of 9. A series, when resorted 

 to as a definition is most convenient if always converging, but 

 in development a series is not to be considered as correct and 

 safe merely on account of its convergence, for expressions may- 

 be assigned which are developable by some incomplete me- 

 thods in a converging series, and yet may be shown, from the 

 functional properties which constitute their best definition, to 

 be e(|ual to n terms of such series together with a remainder 

 (sometimes represcntable in the form of a definite integral,) 

 which, in certain cases, instead of approaching towards as a 

 2 R2 



