This paper presents a mathematical derivation using the classical theory of fluid dynamics for the force on an arbitrarily shaped body in a linear shear flow. To make the analysis tractable, the problem is linearized by assuming that the strain rate is weak and neglecting terms of the order of the strain rate squared. The argument generalizes previous established analytical results due to Darwin regarding the drift-volume and Lighthill for the asymptotic form of the rotational velocity field induced by the body. The final expression for the force is determined by generalizing an analytical argument due to Auton for the sphere. The results identify for the first time a rotational lift force component that occurs only when the body shape is truly asymmetric.