The notion of regularity for {q(n --1) + 1;n}-arcs of a finite projective plane, discussed previously ([3]), is extended to the {q(n --1) + m;n}-arcs of the plane. Following this, conditions for the completeness of regular {q(n --1) + m;n}-arcs are determined.In a recent paper (see [3]) we dealt with the completeness of regular LS-arcs of a projective plane n (q), a regular LS-arc being defined as a {q (n-1) + + 1 ; n}-arc such that through each of its points all the secants of order less than n have the same order (this order can depend on the point)1.The notion of regularity can be extended in a natural way to any (k, n)-arc.Precisely we define a regular point of type i of a {q (n-1) + m; n}-arc C to be a point of C through which there are only secants of order n and order i (1 <~i<<.n-1). In addition to this we say that C is regular if all its points are regular.In this paper we deal with the completeness of regular {q (n-1) + m; n}-arcs, where m >i 2.The following symbols will occur frequently: rf=number of /-secants (1 <<.i<~n) through a point P of the arc; h~=number of/-secants (O<~i<<.n) through a point Q not on the arc; H~=total number of/-secants (0~
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