Mr. A. Cayley on the Theory of Logarithms. 



357 



true generally in whichever quadrant A is situated, i. e. we haVe 

 always E = 0, except in the cases in which the finite line AA' 

 meets the negative portion of the axis of x. But when this 

 happens, then if the line AA', considered as drawn from A to A', 

 passes from above to below the axis of x, we have E = + 2 ; but 

 if the line AA', considered as drawn from A to A', passes from 

 below to above the axis of x, then E= — 2. So that treating 

 the points A, A' as the geometrical representations of the com- 

 plex numbers x+yi, x' -^y'ij we have in an exceedingly simple 

 form the precise determination of the discontinuous number 

 E(=0 or +2) in the formula ' ' % ^ "^ ' ; ^^ 



,DYm-gt,ix log^^ =log(^'+yi)-log(^+2^?y+E7re. ^^-^'^^^ 

 V^ Consider in general the definite integral;'^' ^ ± — ^'^ ^^S" '^? 



^^^ bonis' 





--: f +'=r^ noifj 



^97rJi?go0 ^ 





rrc QjyuaH 



where ^i'', ,^'are compilex num- 

 bers of the forma? + yi,s;' + y'i', 

 and take A, A' as the geome- 

 trical representations of these 

 limits, and the variable point 

 P as the geometrical repre- 

 sentation of the complex vari- „ 

 able u. The value of the 

 definite integral will depend 

 to a certain extent on the 

 series of values which we sup- 

 pose u successively to assume ^ 

 in passing from z to z', or 

 what is the same thing, on;r9ii? 

 the path of the variable point 

 P from A to A'. For (excluding altogether the case in which the 

 path passes through a point for which <j>u becomes infinite) it is 

 well known that the value of the definite integral is the same for 

 any two paths which do not include between them a point for 

 which (pu becomes infinite; but when this condition is not 

 satisfied, then the value of the definite integral is not in general 

 the same for the two paths*. In order therefore to give a pre- 

 cise signification to the notations, we must fix the path of the 

 point P, and it is natural to assume that the path is a right line 



;0=:.i-«i=a 



* The theorem is, I believe, due to M. Cauchy. See the memoir of 

 M. Puiseux, Recherches sur les Fonctions Algebriques^LiowYiWei y9lt xv^ 

 p. 365-480, where the subject is elaborately discussedv^ci^^ Of ,m. lO mii 



