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r 
I XI. On the Foundations of the Theory of Algebraic Functions of One Variable. 
[ By J, C. Fields, Ph.D., Associate Professor of Mathematics in the University 
I of Toronto. 
I Communieated by A. R. Forsyth, Sc.D., LL.D., F.R.S. 
i; 
! Received June 4,—-Read June 6, 1912. 
i § 1. Some years ago the writer published a book* in which he developed a new theory 
of the algebraic functions of a complex variable. The theory in question was purely 
algebraic in its character and perfectly general. The higher singularities gave rise to 
[; no specific difficulties due to their greater complexity and no exceptional cases had 
[ to be reserved for separate treatment. The capital result of the theory might be said 
i to be the “ Complementary Theorem ”—a theorem which is considerably more general 
i than the Riemann-Roch Theorem. 
t The book, however, presents its difficulties for the reader, and, in particular, the 
■ sixth chapter would seem to have been a stumbling-block. For this chapter the 
i writer has already given several comparatively simple substitutes, and the reader of 
[ the present paper will find tliat, among other results, those of the chapter in question 
follow in very easy fashion from the representation of a rational function in the 
form (8). The method of the “ deformation of a product,” which plays a conspicuous 
part in the earlier chapters of the book, is liere dispensed with. The residues of what 
, we call the principal coefficient of the reduced form of a rational function will be 
found to play an important role —a role which is already implied in the argument of 
i the book and which is brought into evidence in a paper by the writer published in 
‘ Vol. XXXII. of the ‘American Journal of Mathematics’ under the title “The 
Complementary Theorem.” In the present paper the apparatus for handling the 
I residues in question will be greatly simplified. We have no need of the function 
R ( 2 , v) defined in Chapter IX. of the book, and at the same time we are able to 
I dispense with the functions ( 2 , v) and the more or less complicated formulse 
; connected with these functions in the earlier presentation of the theory. 
Let 
f{z,u) = + + = 0.. (1) 
* ‘Theory of the Algebraic Functions of a Complex Variable,’ Mayer and Muller, Berlin, 1906. 
) VOL, CCXII. -A 494. 2 x 2 Published separately, January 29, 1913. 
