Many of the rays on circular optical fibres that are predicted from geometric optics to be trapped by total internal reflection are, in fact, leaky. Thus geometric optics fails for long fibres. The letter presents a concise analytic expression for the loss of all weakly attenuating rays. The method is simple, and applicable to a large class of problems. The results provide a unifying theory of light transmission within fibres, representing a generalisation of Fresnel's and Snell's laws.