PREFACE. THE main object of the present treatise is to supply a text-book on Geometrical Optics to students reading for the Mathematical Tripos at Cambridge, who do not wish to proceed much beyond those portions of the subject which are required for the first part of the Tripos Ex amination. The investigations are therefore not carried beyond first approximations. The discussion of the position of the foci of obliquely incident pencils has, however, been brought within this boundary, instead of being derived from the second approximations for direct pencils. The Author hopes that the book may be useful to a wider class of students, not residing in any University, by giving to them a concise view of the mathematical explanation of instruments, with the practical details of which they are familiar. The Author wishes to express his acknowledgements to several friends, for hints and suggestions, and especially to Mr W. M. Spence, Fellow of Pembroke College, Cambridge, for his valuable assistance in revising the book as it went through the press. COLLEGE OF PHYSICAL SCIENCE, The Author is indebted to Professor Hathornthwaite, Elphinstone College, Bombay, for several corrections in the present edition. UNIVERSITY COLLEGE, AUCKLAND, NEW ZEALAND, June, 1893. AN ELEMENTARY TREATISE ON GEOMETRICAL OPTICS. CHAPTER I. LAWS OF REFLECTION AND REFRACTION. 1. THE subject of Optics divides itself naturally into two distinct parts. One of these consists in the deduction by geometrical or analytical methods of the consequences of a few well ascertained laws which govern the simplest phenomena of light. The second consists in the explanation of the mechanical or physical causes which produce those phenomena. These two branches of the subject are usually known as Geometrical and Physical Optics respectively, and it is with the former exclusively that the present treatise is concerned. We shall not discuss the physical causes of the propagation of light, but taking certain laws for granted, we shall endeavour to trace out some of their more interesting and useful consequences. It will be necessary to commence with a few important definitions and explanations. 2. When we are in a place exposed either to the light of the sun or any artificial source of light, we are sensible of the existence of objects surrounding us. If the light of the sun be excluded or the artificial light extinguished, we A. G. O. 1 to us. cease to be able to perceive by sight anything that is near Such bodies as the sun or a lighted lamp have therefore the property of rendering us sensible by sight not only of their own existence, but of that of all other bodies on which they shed what we call their light. Bodies which have this power are called self-luminous bodies. On the other hand, bodies which require the presence of some selfluminous body in order to enable us to see them, are called non-luminous or dark bodies. 3. We assume that the sensation of sight is produced by something (not necessarily material) which comes from the thing seen and enters the eye. Experiment shows that it proceeds in straight lines. We assume farther that this something, which we shall in future call light, proceeds to the eye from every material point of any body which is seen. The quantity of light which proceeds from any material point of a body to the eye we shall call a pencil of light. We shall also suppose that the form of this pencil is a cone, whose vertex is the luminous point, and whose base is the portion of the eye which admits light. 4. If we suppose the vertical angle of this cone to be indefinitely diminished, we get a certain quantity of light which may be considered as a straight line, and is called a ray. It is not necessary for our purposes that such a small quantity of light shall be actually able to exist separately, but it is evident that we may suppose the pencils we have before considered to consist of an indefinite number of small portions, such as we have defined as rays. We may then give the following definitions: (1) A pencil of light is the portion of light, by means of which a given material point of any object might be seen by an eye suitably placed. It is generally considered to be of a conical form with the material point at its vertex. If the material point be at an indefinitely great distance, the cone will become a cylinder. (2) A ray of light is the limiting form of a pencil of light when the solid angle at the vertex of the cone is indefinitely diminished. It is usually considered to be a line; and in accordance with a remark previously made, it is a straight line as long as it continues in the same medium. A pencil is conceived to be made up of an infinite number of such rays. 5. We know by experience that if there be nothing but air or vacuum between us and any luminous object, the light of that object is able to reach our eyes. If we interpose a piece of glass, or ice, or a rectangular vessel of glass containing clear water between our eyes and the luminous object, the light is still able to produce the sense of sight. If, on the other hand, we hold up a piece of wood or iron between our eyes and the object, the latter becomes invisible to us, the light not being able to traverse the wood or iron. We thus get an optical distinction between different classes of bodies. Some bodies permit light to traverse them more or less freely and regularly: others refuse to allow it to pass at all. Bodies of the former class are called transparent bodies; of the latter, opaque. 6. There are some bodies, as alabaster, porcelain, which, when held up between our eyes and a strongly luminous body, as the sun, allow light to pass in an irregular way, but do not permit us to see the luminous body distinctly. Such bodies are called translucent. Light transmitted through such bodies will not be farther considered in this book, as it obeys no simple geometrical laws. 7. When a body is considered with reference to its power of transmitting light, it is usually called a medium. 8. When a pencil of light proceeding in one medium is incident on the surface of another transparent medium, it is usually divided into three parts. (1) A portion is reflected back into the original medium according to a law to be hereafter stated. (2) A portion passes into the new medium according to another law to be hereafter stated, and is said to be refracted into the new medium. (3) A third portion is employed in rendering visible the surface which separates the two media. |