ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 283 



PROP. IX. It is impossible, by means of any combination of reflexions 

 and refractions, to produce a perfect image of an object at two different distances, 

 unless the instrument be a telescope, and 



l = n = ^- 1 , m=l. 

 P* 



It appears from the investigation of Prop. VIII. that the results there 

 obtained, if true when the objects are very small, will be incorrect when the 

 objects are large, unless 



and it is easy to prove that this cannot be, unless all the lines in the one figure 

 are proportional to the corresponding lines in the other. / 



In this way we might show that we cannot in general have an astigmatic, 

 plane, undistorted image of a plane object. But we can prove that we cannot 

 get perfectly focussed images of an object in two positions, even at the expense 

 of curvature and distortion. 



We shall first prove that if two objects have perfect images, the reduced 

 path of the ray joining any given points of the two objects is equal to that 

 of the ray joining the corresponding points of the images. 



Let a 2 (fig. 8) be the perfect image of a t and & of &. Let 



Draw a, l D l parallel to the axis to meet the plane JB lf and aj) 3 to the plane 

 of B v 



Since everything is symmetrical about the axis of the instrument we shall 

 have the angles D l B 1 ^ 1 = D t Bj3, = 0, then in either figure, omitting the suffixes, 



6. 



It has been shown in Prop. VIII. that the difference of the reduced paths 

 of the rays ,& a^ in the object must be equal to the difference of the reduced 

 paths of a 2 6 2 , a^ in the image. Therefore, since we may assume any value for 6 



Pi /(." + 61* + GI - 2A cos 0) - p., J(a* + 6 2 2 + c 2 " - 2a 2 6 2 cos 6} 



362 



