The following problems are considered:I. Paraxial ray tracing by the focal plane method.—For spherical refracting and reflecting surfaces it is shown that by drawing two secondary axes, one parallel to a ray incident at an arbitrary angle and the other parallel to the path of the same ray after refraction or reflection, the positions of the focal planes are readily established and their necessary properties deduced. The same is done for a lens by a method based on the fact that a focal plane of the first surface is conjugate to a focal plane of the second surface.II. The variation in the distance L between the conjugate foci of a thin lens (focal length f) with the distance l of one focus from the lens.—A general formula is established, and a simple construction for plotting the graphs betweenL/fandl/fis described.III. Refraction by prism combinations and the prism refractometer.—A graphical method is used to discuss the design of the Amici prism, the direct vision spectroscope, and achromatic prisms giving appreciable deviations. A graphical method is also described for finding the refractive index of a liquid from results obtained with a prism refractometer.