An index formula is proved for elliptic systems of P.D.E.'s with bound
ary values in a simply connected region Omega in the plane. Let A deno
te the elliptic operator and B the boundary operator. In an earlier pa
per by the author, the algebraic condition for the Fredholm property,
i.e. the Lopatinskii condition, was reformulated as follows. On the bo
undary, a square matrix function Delta(B)(+) defined on the unit cotan
gent bundle of partial derivative Omega was constructed from the princ
ipal symbols of the coefficients of the boundary operator and a spectr
al pair for the family of matrix polynomials associated with the princ
ipal symbol of the elliptic operator. The Lopatinskii condition is equ
ivalent to the condition that the function at Delta(B)(+) have inverti
ble values. In the present paper, the index of (A, B) is expressed in
terms of the winding number of the determinant of Delta(B)(+).