The linear stability of the Bickley jet in the framework of the beta-plane approximation is considered, with the objective of presenting analytical calculations which add to previous numerical computations. It is well-known that the equation governing the neutral solutions which are analytic at the critical layer can be transformed into an associated Legendre equation. It turns out that this particular equation has simple closed-form solutions other than those known already, which are the Legendre polynomial of degree two, and two associated Legendre functions of the first kind, respectively. This observation makes it possible to find analytic neutral modes and corresponding neutral curves in the $(\beta,k)$-plane not known previously, both for the bounded and the unbounded Bickley jet. Here $\beta$ denotes the beta-parameter and $k$ the wavenumber. These neutral curves comprise parts of the stability boundary. It is shown that the line segment ($\beta=-2$, $0<k<\sqrt{2}$) is part of the stability boundary for the unbounded Bickley jet, so the region for the unstable radiating modes is larger than the one obtained previously. Also, analytic sinuous and varicose modes and corresponding neutral curves are found in the case of the bounded flow where only numerical calculations have previously been presented. Furthermore, local stability analysis reveals weakly amplified modes with wave speed outside the range of the velocity profile for the Bickley jet. This is rather rare, although Pedlosky's theorem allows for it, and there are only a few examples of flows in which such modes occur. Here these modes are sinuous modes and occur when the flow is both bounded and unbounded.