6 KEVIN W. 0. KADELL

(2.2)

= I

i = 1 X *

aiSP

i=1 *

Ki*

The extension is uniquely determined by applying (2.2) to the

representation (2.1) of P in terms of elemental paths. It is linear

since it satisfies

(2.3) S(.p) = -Sp and S ^ ^ p ^ = Sp^

+

Sp^, e(P,) = s(P2),

which is equivalent to (2.2).

Our path functions satisfy an analog of the Cauchy integral theorem.

We have the fundamental relation

(2.*)

Vi,j +Hi+1,J

H, . + V. . „,

Consider the unit square with upper left corner (i,j),

go from (i,j) to (i+1,j+1) are given below.

Two paths which

d , j )

(2.5)

U + 1.J)

(l.J+D

(i+1,J+1)

By (2.3), SRi = Vi(.

+

Hl+1jJ and S^ = ^ ^ • ^ J+1. Thus (2..)

becomes Sp = Sp . If we are going to "integrate from (i,j) to

K1 K2

(i+1,j+1)," then we are at liberty to select P1 or P as the "path of

integration." The fundamental relation (2.4-) is the local analog of the

Cauchy integral theorem for the square (2.5). The global version is given

by the following lemma.

Lemma 1. Let the path function Sp be linear (2.2). If the

fundamental relation (2.4) holds, then Sp only depends on s(P) and

e(P).