360 Mr. A. Cayley on the Theory of Logarithms, 



where the new variable r is real. The two integrals in r are 

 identically the same^ and consequently we have in every case 





1 . . y 



Now log' tt= — ; and in passing from m = 1 to ti= — , there is no 

 u z 



discontinuity in the value of log w 



= log >/^M^ + i(^tan-' |- + €7rj, , ^ji jno- , 



if for the moment u-=.'p'\-qi ; hence the value of the integral on 



the left-hand side is simply log - . The value of the integral on 



the right-hand side is in like manner, log s' — log z in the case 

 in which the finite right line from u-=.z to u^z^ does not meet 

 the negative part of the axis of x ; but when this happens, then 

 there is a discontinuity in the value of the logarithm, and the 

 integral on the right-hand side will be log,?'— log;?— 27ri, or 

 log^' — log2' + 27r2, according as the right line considered as 

 drawn from z to ^ passes from below to above or from above to 

 below the negative part of the axis of x. We have therefore in 

 every case (E being defined as above) log-sr' — logs' + ETr? forthe 

 value of the integral on the right-hand side, and the relation 

 between the two integrals gives, as it ought to do, the equation 



log — = log 2^^ — log sr -f- E7ri, 

 z 



or in the form in which it was before written, 



^°^^+^* =log(a^+ya)-log(a?-f-yO + E7n. 



The preceding discussion shows that the discontinuity in the 

 value of E(— or +2) arises from or is most intimately con- 

 nected with the geometrical discontinuity which necessarily 



exists in the surface x=tan~^~, whenever we define the sym- 



X 



bol tan~' in such manner as to give a unique value to the coor- 

 dinate z. 



Stone Buildings, 

 Sept. 19, 1856. 



