INTRODUCTION. 5 



familiar construction, by means of which the position and magni- 

 tude of the image of a given object can be easily determined. Let 

 two lines p r and p q be drawn from the extremity p (Fig. 3) of 

 the object p t, one of which cuts the axis in F t whilst the other 

 proceeds parallel to it. From q and r, the points of intersection 

 with the plane E, let two other lines r p* and q p* be drawn, 

 whose directions are, as it were, interchanged with the former 

 lines, r p* being parallel to the axis, and q p* intersecting it at 

 .F*. It is evident that the two lines p r and p q represent two 



FIG. 3. 



rays, which, after refraction, unite at ^>*, and as all the other rays 

 which proceed from p are refracted in a like manner to p*, the 

 image of the point p is formed at p*. By a similar construction 

 the image of the other extremity t, as well as of all other points 

 between p and t, may be obtained. To simplify the matter still 

 further we can replace one of the two lines by the ray of direction, 

 p E p* 9 which passes through unrefracted. 



If we would extend the terminology introduced by Gauss to 

 this simple case of an infinitely-thin lens, the planes F and F* 

 would be termed the focal planes, and the plane E the principal 

 plane. The distances of the focal planes from the principal plane 

 represent the focal distances, and their points of intersection with 

 the axis, the focal points. 



It must, however, be borne in mind that the above construction, 

 as well as the formulae generally given in the text-books of 

 Physics, by which the position of the image is calculated, are only 

 accurate, even for infinitely-thin lenses, on the two suppositions 

 that the rays from the object form very small angles with the 

 axis, and that the effective portion of the refracting surface is only 

 a small part of the whole spherical surface. These are limitations 



