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



(of course there are an infinity of paths which give the same value 

 to the definite integral, or as we may call them, paths equivalent 

 to the right line ; but the consideration of these would be a need- 

 less complication of the definition, and it is better to attend to 

 the single path — the right line). The definition is at once 

 converted into an analytical one; we have only to assume 

 t«=^H-r(x:'— r), and to suppose that the new variable r passes 

 from r=0 (which gives u:=z) through real values tor = l (which 

 gives u-s^s^), i. tf. we have as the equivalent analytical definition 

 of the definite integral between the complex limits e, ^ the 

 equation 



j <t>udu=(sf'^x)\ </)[<8r+r(5r'— j8:)](fr, 



where the new variable r is real. The only restriction is, that 

 <f>u must not become infinite for any value of u along the path 

 in question, i, e, <f>\z '{'r{s^ —zy] must not become infinite for any 

 real value of r between the limits r=0, r= 1. 



Suppose next, the path being defined as above, or in any 

 other manner, that <f>iu is a function of u such that (j>lu—(f>u. 

 Then if <f>^u is continuous along the entire path, we have 



i 





but if <f>iu is discontinuous at any points of the path, e. g. at the 

 point M=Wp and at no other point, then 



t' 



(f>u du = <f>f^ — (f>i{u^ + a') + <^>^{u^ — a) — tf)^, 



where Uj—af, Ui-^a! are values indefinitely near to «,, the path 

 being from z through u^—a to Uj-i-a! and thence to z'. Or if 

 we represent the break 0/Wy + a')— <^^(wy-f «) by the symbol V, 

 then we have 



Suppose now 



f=tan-*-^ + c7r, 



where, as before, tan""* — denotes an arc between the limits 



, H — , and , a 



07=—, €=±l=y', 



and to fix the ideas, consider ^ as the xr-coordinate of a surface^ 



