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THE ROYAL SOCIETY OF CANADA 



slightly sepaiated, to show that the complete contoui can be described 

 continuously by a point moving in the sense of the arrows. 



In the first figure the banks of A, B are coincident; the arrows in 

 figure 1 correspond to the arrows along the outer banks in fig. 2. Hav- 

 ing made this choice of arrows, the outer banks of A, B in fig. 2 furnish 

 the senses of the arrows for fig. 1, and with reference to them, we call 

 the outer banks of A, B the left banks, and the inner banks of A, B 

 the right banks. 



Figure 1 



Figure 2 



For many purposes it is desirable to contract A, B till A lies in 

 its limiting position along the branch-cut a^ aj, and B becomes an 

 infinitely thin oval, coincident with a line from a 2 to a^. 



The integral of the first kind 





dx 



y 



is nowhere infinite on T^ and is one-valued, that is, its value at a place 

 (x, y) on T^ is independent of the path from the initial place (x^, y„). 

 Along A, B the values on opposite banks are unequal, but the difference of 

 these values is constant along a cross-cut. For instance Uj^-u^ = Up-Uq 

 = Uj-Ug = / du taken on T^ from q to p or from s to r. 



The values at opposite points of the left and right 

 banks of A, B may be denoted by u,, u^; the difference u,-Uj 

 along a cross-cut is called the modulus of periodicity of the integral 

 at the cross-cut. Denote these moduli of periodicity at A and B by 

 2û>, 2w'. Then the modulus of periodicity at B is equal to the value 

 of the integral / du round A in the sense of the arrow and the modulus 

 of periodicity at A is equal to the value of / du taken round B in the 

 sense opposite to that of the arrow, where the arrows are those in 

 Fig. 1. 



Hitherto we have been considering T', not T. On T suppose 

 that A is crossed from left bank to right bank by the path of integration. 

 The value of the integral on the right bank is now not u, but u, = 

 Ur + (u,-Ur) =Uj + 2w. This shows that the effect of m crossings of A 



