274 ON THE GENERAL LAWS OF OPTICAL INSTRUMENTS. 



are parallel, which is a particular case of a perfect pencil. So that for every 

 position of the focus of incident rays, the emergent pencil is free from astig- 

 matism. 



PROP. II. In an instrument, perfect at two different distances, the image 

 of any plane object perpendicular to the axis will be free from the defects of 

 curvature and distortion. 



Through the point P l of the object draw any line P& in the plane of 

 the object, and through P,Q, draw a plane cutting the planes A lt -B, in the lines 

 0,0,, 6j8,. These lines will be parallel to P,<?, and to each other, wherefore 

 also their images, 0,0,, &^8,, will be parallel to P l Q l and to each other, and 

 therefore in one plane. 



Now suppose another plane drawn through P,<?, cutting the planes A l and 

 BI in two other lines parallel to PjQ^ These will have parallel images in the 

 planes A, and B t , and the intersection of the planes passing through the two 

 pairs of images will define the line P t Q t which will be parallel to them, and 

 therefore to P&, and will be the image of P,(?,. Therefore P t Q t , the image 

 of PjQj is parallel to it, and therefore in a plane perpendicular to the axis. 

 Now if all corresponding lines in any two figures be parallel, however the lines 

 be drawn, the figures are similar, and similarly situated. 



From these two propositions it follows that an instrument giving a perfect 

 image at two different distances will give a perfect image at all distances. We 

 have now only to determine the simplest method of finding the position and 

 magnitude of the image, remembering that wherever two rays of a pencil inter- 

 sect, all other rays of the pencil must meet, and that all parts of a plane 

 object have their images in the same plane, and equally magnified or diminished. 



PROP. III. A ray is incident on a perfect instrument parallel to the axis, 

 to find its direction after emergence. 



Let a,6, (fig. 2) be the incident ray, -*!,, one of the planes at which an 

 object has been ascertained to have a perfect image. Afi^ that image, similar 

 to A l a 1 but in magnitude such that Aji t = xA l a l . 



Similarly let B t b t be the image of Bfa, and let B^ yB^. Also let 

 -4,-B^c, and A t B i = c i . 



Then since a, and b t are the images of a, and & the line F0Jb t will be 

 the direction of the ray after emergence, cutting the axis in F t , (unless x = y, 



