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THEORY OF ALGEBRAIC FUNCTIONS OF ONE VARIABLE. 373 
value z = a^. To see this it is only necessary to remember that the aggregate sum of 
all the orders of coincidence of any rational function is equal to 0 and to note that 
the general rational function built on the basis (f) defined with reference to the 
basis (t) by the relations (84) and (85) is obtained on multiplying by V\,{z,u)lf^'{z,'>i), 
the general rational function built on the basis originally defined as complementary 
to the basis (r). 
While the complementary theorem has here been deduced on the hypothesis that 
the fundamental equation is an integral algebraic equation, it is easy to verify^that 
the generalized theorem holds also when the^ fundamental equation is not integral. 
For more detail in connection with the theorem, and for some of its consequences, the 
reader is referred to Chapter XII. and the following chapters of the hook already 
cited. In the present paper it has been the object of the writer to present a more 
simple and elegant treatment of the theory leading up to the complementary theorem. 
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