The circle contractivity condition is essentially a particular form of stability condition, to be obeyed when a linear multistep method is applied to an initial-value problem involving the differential equation \((d/dt)y=\lambda (t)y\), for some continuous function \(\lambda\) (t). The author demonstrates the existence of a class of such methods, and lists formulae of order 1 to 12. An indication is given that their absolute stability regions may be somewhat better than those for the corresponding Adams-Moulton formulae for the higher orders.