284 ON THE CKNKUAL LAWS OF OPTICAL INSTRUMENTS. 



is constant for all values of 6. This can be only when 



fh x/(,' + V + c,') = fr s/K + 6,' + c,'), 

 and 



which shows that the constant must vanish, and that the lengths of lines 

 joining corresponding points of the objects and of the images must be inversely 

 as the indices of refraction before incidence and after emergence. 



Next let ABC, DEF (fig. 9) represent three points in the one object 



and three points in the other object, the figure being drawn to a scale so that 



all the lines in the figure are the actual lines multiplied by /*,. The lines of 



the figure represent the reduced paths of the rays between the corresponding 

 points of the objects. 



Now it may be shown that the form of this figure cannot be altered with- 



out altering the length of one or more of the nine lines joining the points ABC 



to DEF. Therefore since the reduced paths of the rays in the image are equal 



to those in the object, the figure must represent the image on a scale of fj. 

 to 1, and therefore the instrument must magnify every part of the object alike 



and elongate the distances parallel ,to the axis in the same proportion. It is 

 therefore a telescope, and m=l. 



If fr = fa, the image is exactly equal to the object, which is the case in 

 reflexion in a plane mirror, which we know to be a perfect instrument for all 

 distances. 



The only case in which by refraction at a single surface we can get a 

 perfect image of more than one point of the object, is when the refracting 

 surface is a sphere, radius r, index \L, and when the two objects are spherical 



7* 



surfaces, concentric with the sphere, their radii being - , and r ; and the two 

 images also concentric spheres, radii p.r, and r. 



In this latter case the image is perfect, only at these particular distances 

 and not generally. 



I am not aware of any other case in which a perfect image of an object 

 can be formed, the rays being straight before they enter, and after they emerge 

 from the instrument. The only case in which perfect astigmatism for all pencils 

 has hitherto been proved to exist, was suggested to me by the consideration 



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