PRELIMINARY DIOPTRIC CONSIDERATIONS. 



827 



each other and to the axis O Q (from a to a 5 ) ; and the rays coming from r run 

 parallel to each other in the first medium, but not parallel to the axis O Q (from 

 c to c 2 ). (3) All rays in the second medium that are parallel to each other (from 

 b to bg) and to the axis O Q are united at a point in the first medium (p) , the 

 first principal focus (the converse of this proposition is also true) . A plane erected 

 at this point perpendicular to O Q is known as the first focal plane (A B). The 

 radius (m x) of the refracting surface is equal to the difference between the dis- 

 tances of the principal focal points (p and p t ) from the principal point (x) ; hence 

 mx = pi x p x. 



FIG. 276. 



1. From a knowledge of these simple relations the direction of the refracted 

 ray may be constructed. Let A (Fig. 276) be the first, B the second medium; 

 c d the spherical surface between them; a b the optical axis, k the nodal point, p 

 the first and pj the second principal focus; C D the second focal plane. If, now, 

 x y is the direction of the incident ray, what is the direction of the refracted ray 

 in the second medium ? 



Construction: Draw the undeviated axial ray P k Q parallel to x y. Then the 

 line y Q must be the direction of the refracted ray (according to proposition 2) . 



2. Construction of the Image of a Given Point in an Object. [The letters A, B, 

 c d, a b, k, p and p lf C D,in Fig. 277 have the same designations as before.] If 

 now a point of light be given at o, where will be its image in the second medium? 



Construction: Draw the undeflected axial ray o k P. Then draw the ray o x par- 

 allel to the axis a b. The parallel rays a e and o x are united at p t (according to 



FIG. 277. 



proposition i). If x p t is prolonged until it intersects the ray o P, P is the image 

 of the point o, for the image will be situated at the intersection of the rays o x and 

 o k in the second medium, consequently at P. 



Construction of the Refracted Ray and of the Image when Several Refractive 

 Media are Present. If several refractive media are placed behind one another, 

 the construction must be made from medium to medium in the manner already 

 described. This, however, would be a troublesome procedure, especially in 

 dealing with small objects. In 1840 Gauss calculated (by methods that cannot 

 be explained in an elementary treatise) that in all such cases the method of con- 

 struction may be greatly simplified. If the media are "centered," that is if all 



