This paper by P. G. Saffman investigates the lift force on a small sphere moving through a viscous fluid in a slow shear flow. The sphere moves with velocity V parallel to the streamlines, and the lift force is found to be perpendicular to the flow direction. The force is proportional to $ a^2 \kappa^{1/2} / \nu^{1/2} $, where $ a $ is the sphere radius, $ \kappa $ is the velocity gradient, and $ \nu $ is the kinematic viscosity. The result is relevant to observations by Segré & Silberberg (1962) of small spheres in Poiseuille flow, where spheres migrate laterally in a tube. The paper also discusses the problem of a sphere in a parabolic velocity profile and the dependence of the lift force on various parameters. The analysis considers both inner and outer expansions of the flow field. The inner expansion is used to calculate the force on the sphere due to the first-order term in the velocity field, while the outer expansion is used to determine the dominant terms in the force. The lift force is found to be proportional to $ a^2 \kappa^{1/2} / \nu^{1/2} $, with a numerical coefficient of approximately 81.2. The result is validated by comparing it with the observations of Segré & Silberberg, and it is shown that the lift force is independent of the particle's rotation to leading order. The paper also discusses the effect of inertia on the motion of a sphere in a shear flow, and the importance of the tube walls in generating the lateral force. The results are applied to the case of a sphere in a parabolic velocity profile, and the dependence of the lift force on the various parameters is evaluated. The paper concludes that the lift force is primarily due to the shear flow and is independent of the particle's rotation to leading order. The results are consistent with observations of particle migration in Poiseuille flow and provide a theoretical basis for understanding the phenomenon.This paper by P. G. Saffman investigates the lift force on a small sphere moving through a viscous fluid in a slow shear flow. The sphere moves with velocity V parallel to the streamlines, and the lift force is found to be perpendicular to the flow direction. The force is proportional to $ a^2 \kappa^{1/2} / \nu^{1/2} $, where $ a $ is the sphere radius, $ \kappa $ is the velocity gradient, and $ \nu $ is the kinematic viscosity. The result is relevant to observations by Segré & Silberberg (1962) of small spheres in Poiseuille flow, where spheres migrate laterally in a tube. The paper also discusses the problem of a sphere in a parabolic velocity profile and the dependence of the lift force on various parameters. The analysis considers both inner and outer expansions of the flow field. The inner expansion is used to calculate the force on the sphere due to the first-order term in the velocity field, while the outer expansion is used to determine the dominant terms in the force. The lift force is found to be proportional to $ a^2 \kappa^{1/2} / \nu^{1/2} $, with a numerical coefficient of approximately 81.2. The result is validated by comparing it with the observations of Segré & Silberberg, and it is shown that the lift force is independent of the particle's rotation to leading order. The paper also discusses the effect of inertia on the motion of a sphere in a shear flow, and the importance of the tube walls in generating the lateral force. The results are applied to the case of a sphere in a parabolic velocity profile, and the dependence of the lift force on the various parameters is evaluated. The paper concludes that the lift force is primarily due to the shear flow and is independent of the particle's rotation to leading order. The results are consistent with observations of particle migration in Poiseuille flow and provide a theoretical basis for understanding the phenomenon.