280 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 



4, 1, 4 and the distances 4 and 4. This combination simply inverts every object 

 without altering its magnitude or distance along the axis. 



The preceding theory of perfect instruments is quite independent of the 

 mode in which the course of the rays is changed within the instrument, as 

 we are supposed to know only that the path of every ray is straight before 

 it enters, and after it emerges from the instrument. We have now to con- 

 sider, how far these results can be applied to actual instruments, in which 

 the course of the rays is changed by reflexion or refraction. We know that 

 such instruments may be made so as to fulfil approximately the conditions of 

 a perfect instrument, but that absolute perfection has not yet been obtained. 

 Let us inquire whether any additional general law of optical instruments can 

 be deduced from the laws of reflexion and refraction, and whether the imper- 

 fection of instruments is necessary or removeable. 



The following theorem is a necessary consequence of the known laws of 

 reflexion and refraction, whatever theory we adopt. 



If we multiply the length of the parts of a ray which are in different 

 media by the indices of refraction of those media, and call the sum of these 

 products the reduced path of the ray, then : 



I. The extremities of all rays from a given origin, which have the same 

 reduced path, lie in a surface normal to those rays. 



II. When a pencil of rays is brought to a focus, the reduced path from 

 the origin to the focus is the same for every ray of the pencil. 



In the undulatory theory, the " reduced path " of a ray is the distance 

 through which light would travel in space, during the time which the ray 

 takes to traverse the various media, and the surface of equal "reduced paths" 

 is the wave-surface. In extraordinary refraction the wave-surface is not always 

 normal to the ray, but the other parts of the proposition are true in this and all 

 other cases. 



From this general theorem in optics we may deduce the following propo- 

 sitions, true for all instruments depending on refraction and reflexion. 



- PROP. VIII. In any optical instrument depending on refraction or reflex- 

 ion, if 0,0,, &,/?, (fig. 7) be two objects and 0,0,, &j/3, their images, A l B l the 

 distance of the objects, A t B t that of the images, /x, the index of refraction of 



