Let Q _q_ be a section of a lens, and _f_ A _r_ its optical axis, or the line in which a ray of light passes unchanged in its direction through the lens, from its being normal to both surfaces, whether the lens be double-convex as above, or plano-convex (see fig. 53), then the _principal focus f_ is that point where the rays from _r r r_, which fall parallel to the optic axis on the outer face of the lens, meet after refraction at the two faces,--or, to speak more in the language of the art which is under consideration, the _principal focus f_ is the point whence the rays of light, proceeding in their naturally divergent course, fall on the inner surface Q A _q_ of the lens, and are so changed by refraction there and at the outer face, that they finally emerge parallel to the _optic axis_ in the directions Q _r_, _q r_. The position of this point depends partly on the refractive power of the substance of which the lens is composed and partly on the curvature of the surface or surfaces which bound it. It would be quite beyond the scope of these Notes to attempt to present the subject of refraction at spherical surfaces before the reader’s view in a rigorous or systematic manner, and thus to advance, step by step, to the practical application of refracting instruments, as a means of directing and economising the light in a Pharos. This would involve the repetition, in a less elegant form, of what is to be found in all the works on optics; and instead of this, I am content to refer, where needful, to those works, and shall confine myself simply to what concerns Lighthouse lenses and their use. It would also be superfluous to determine the position of the principal focus of a plano-convex lens, in terms of the refractive index and radius of curvature,[55] as it can be very accurately found in practice by exposing the instrument to the sun, in such a manner that his rays may fall upon it in a direction parallel to its axis. The point of union between the converging and diverging cones of rays (where the spectrum is smallest and brightest), which is the _principal focus_, is easily found by moving a screen behind the lens, farther from or nearer to it as may be required. The path of the Lighthouse optician, moreover, generally lies in the opposite direction; and his duty is not so much to find the focal distance of a ready-made lens, as to find the best form of a lens for the various circumstances of a particular Pharos, whose diameter, in some measure, determines the focal distance of the instruments to be employed. All, however, that I shall really have to do is to give an account of what has been done by the late illustrious FRESNEL, who seems to have devoted such minute attention to every detail of the Dioptric apparatus, that he has foreseen and provided for every case that occurs in the practice of Lighthouse illumination. His brother, Mons. LEONOR FRESNEL, who succeeded him in the charge of