ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 273 



An image free from these three defects is said to be perfect. 



In Fig. 1, p. 285, let A^a^ represent a plane object perpendicular to the 

 axis of an instrument represented by I., then if the instrument is perfect, as 

 regards an object at that distance, an image A^a^ will be formed by the 

 emergent rays, which will have the following properties : 



I. Every ray, which passes through a point a, of the object, will pass 

 through the corresponding point a 2 of the image. 



II. Every point of the image will lie in a plane perpendicular to the axis. 



III. The figure A.fi.fi, will be similar and similarly situated to the figure 



Now let us assume that the instrument is also perfect as regards an object 



in the plane P-,&A perpendicular to the axis through B u and that the image 



of such an object is in the plane -B a 6jj8 3 and similar to the object, and we 

 shall be able to prove the following proposition : 



PROP. I. If an instrument give a perfect image of a plane object at two 

 different distances from the instrument, all incident rays having a common focus 

 will have a common focus after emergence. 



Let P, be the focus of incident rays. Let P,oA be any incident ray. 

 Then, since every ray which passes through a, passes through a a , its image after 

 emergence, and since every ray which passes through 6, passes through b,, the 

 direction of the ray P,a,&, after emergence must be ajb t . 



Similarly, since a, and /?, are the images of a, and /?,, if P,a,j8, be any 

 other ray, its direction after emergence will be cg3 3 . 



Join a,a,, 6,/J,, c^a,, b^ 3 ; then, since the parallel planes Aja,^ and Bf>^ 

 are cut by the plane of the two rays through P,, the intersections e^o, and 

 frjS, are parallel. 



Also, their images, being similarly situated, are parallel to them, therefore 

 a,o, is parallel to fc-jS,, and the lines a a b, and a^ are in the same plane, and 

 therefore either meet in a point P 3 or are parallel. 



Now take a third ray through P,, not in the plane of the two former. 

 After emergence it must either cut both, or be parallel to them. If it cuts 

 both it must pass through the point P 2 , and then every other ray must pass 

 through P,, for no line can intersect three lines, not in one plane, without 

 passing through their point of intersection. If not, then all the emergent rays 



VOL. I. 35 



