[From the Proceedings of the London Mathematical Society, Vol. iv.] 



LVII. On the Focal Lines of a, Refracted Pencil. 

 [Read April 10th, 1873.] 



HAMILTON has shewn, by means of his Characteristic Function, that in 

 wliatever manner a pencil may be refracted, the rays always pass through t\v< 

 focal lines, the planes through which and the axis of the pencil are at right 

 angles to each other. The same method leads to the following geometrical 

 construction for finding the focal lines of any thin pencil after refraction through 

 a curved surface, the focal lines of the incident pencil and the nature of the 

 curvature of the surface being given. 



1. The characteristic function of a thin pencil whose axis coincides with 

 the axis of z, and whose rays pass through focal lines in the planes of xz and 

 yz at distances a and b from the origin respectively, is for points near the 

 origin 



If we turn this system of rays round the axis of z, through an angle 

 reckoned from x towards y, the expression for V becomes 



(2), 



where 



(3). 



