The first problem is to find the spherically symmetrical distribution of refractive index inside a sphere (the refractive index outside being uniform) that brings rays starting from a given external point A, and falling on the sphere, to an exact focus at a given external point B, where AB necessarily passes through the centre O of the sphere. The solution given by Luneberg (1944) is here derived from an integral equation of Abel's type, in a way slightly different from Luneberg's, and is then developed so as to provide numerical results. Coefficients of general utility in the problem are given in table 1. The remaining tables relate to particular cases, and especially to the three cases in which, A being at infinity so that the incident beam consists of parallel rays, OB equals 2$\cdot $3, 2$\cdot $5 and 2$\cdot $7 times the radius of the sphere. In these three cases the refractive index at the centre of the sphere is respectively 1$\cdot $150, 1$\cdot $137 and 1$\cdot $126 times that outside the sphere, and the variation of refractive index from centre to surface is indicated in final form in table 3. The second problem is to find the axially symmetrical distribution of refractive index in a cylindrical lens terminated by a plane face perpendicular to the axis that brings a parallel beam travelling parallel to the axis and incident normally on the plane face exactly to a focus at a point on the axis of the lens. It is found, again from an integral equation of Abel's type, that the refractive index must be proportional to sech ($\pi $r/2F), where r denotes distance from the axis and F denotes the distance of the focus from the plane face.