[From the Proceedings of the London Mathematical Society, Vol. vi. No. 83.] 



LXXII. On the Application of Hamilton's Characteristic Function to the Theory 

 of an Optical Instrument symmetrical about its axis. 



[Read April 8th, 1875.] 



WHEN a ray of light passes from the point (x lt y lt zj to the point 

 (x v y t , z 2 ) through any series of media, the line-integral V\p,ds may be defined 

 as the distance which light would travel in vacuum in the same tune as it 

 travels from (# y lt z^ to (x 3 , y 3 , z a ). 



Calling this the reduced distance between (x 1} y lt Zj) and (x 2 , y 2 , z 2 ), Hamil- 

 ton's Characteristic Function may be defined as the value of the reduced distance 

 between two points expressed in terms of the co-ordinates of these points. 



It is not necessary that the co-ordinates of the two points should be 

 referred to the same system of axes. In treating of optical instruments we 

 shall reckon z t and z a in opposite directions along the axis, and from different 

 planes of reference. We shall, however, make the axes of x l and a? 2 parallel to 

 each other in what follows. 



Let a ray from the point P i (x l , 0, zj pass through the first plane of 

 reference at the point J2, (a,, & l} 0), through the second plane of reference at 



& 0), and reach the point P 2 (a; 2 , 0, z 2 ); 



then, putting 

 and 



= (x 2 atf+bf+zf^', 





