Resolution of Algebraical Equations. 147 



xr 

 {const. + A . x + B . — + &c. 



+ bA. (x) - - -&e.J. 



x ' 



And the limiting- values of x, which correspond to the former 

 limits of 6 are e", and 6 »+^^r both of which are equal. But the 

 integral does not vanish, though the numerical values of the 

 limits are equal ; for it has been above shewn to be equal to 

 2-xb J -i, which may be easily explained; thus, 



Ax or Ae» between limits = Ae +2w ^'-A€ e 



= Ae 9 .(e i '- J ~ 1 -l)=0. 



. Bx~ 



Similarly — between limits = 0, &c. 



but b\.x between limits = b 1. e e+ * w ~ l -b 1. e" 



= b{e + 2ir s f^\)-bd 



though, therefore, the quantity under the transcendent sign ol 

 log. is the same, at both the limits of the integral ; yet, if we 

 suppose x to circulate through a series of values, until it again 

 arrives at the value from which it set out, we must then make 

 use of the multiplicity of the values of the log. (*), (which are all 

 included under the form <Zmir J~^\ + real log..*) by giving to it. 

 at its limit, that value it acquires by one circulation, namely 

 2t -J -1 +real log. (x). 



It is the same, with other transcendent functions arising from 

 integration, and possessing the character of having an infinity of 

 values in arithmetical progression; as sin"' Or), tan' 1 Or), &c. ; 



T2 



