Let C2p1 be the class of real functions of a real variable that are 2p-periodic and have a continuous derivative. The positive linear operators of the Jackson type are denoted by Ln,p(n ), where p is a fixed positive integer. The object of this paper is to determine the exact degree of approximation when approximating functions f e C2p1 with the operators Ln,p. The value of maxx¦Ln,p(f x) - f(x)¦ is estimated in terms of ¿1(f; d), the modulus of continuity of f', with d = p/n. Exact constants of approximation are obtained for the operators Ln,p (n , p = 2) and for the Fejér operators Ln,1 (n ). Furthermore, the limiting behaviour of these constants is investigated as n ¿ 8, and p ¿ 8, separately or simultaneously.