The two-stream instability for arbitrary but equal temperatures of the streams is investigated. The equilibrium distribution function for a given temperature, which is the sum of two Maxwellian distributions, is approximated by resonance functions . The division of the two-stream instability into convective and non-convective is described. Non-convective instability occurs when the two streams are in opposite directions and the velocity of the slower of the two is greater than a critical value. This critical value is given with considerable accuracy by the empirical formula , where v1 is the speed of the faster stream and u0 the 'thermal' velocity. Comparison is made with a step-function distribution calculation performed by Sturrock.