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

 LXIV. On Hamilton's Characteristic Function for a Narrow Beam of Light. 



[Read January 8th, 1874.] 



HAMILTON'S characteristic function V is an expression for the time of pro- 

 pagation of light from the point whose co-ordinates are x lt y 1} z^ to the point 

 whose co-ordinates are x t , y 3 , z,. It is a function of these six co-ordinates of 

 the two points. The axes to which the co-ordinates are referred may be 

 different for the two points. 



In isotropic media the differential equation of V may be written 



where /i is the slowness of propagation at a point in the medium whose co " 

 ordinates are x, y, z, and is a function of these co-ordinates. If the time of 

 propagation through the unit of length in vacuum be taken as the unit of 

 time, then \L is the index of refraction of the medium. 



The form of the equation in doubly refracting media, as given by Hamilton, 

 is not required for our present purpose. 



Let OPQR be the path of a ray of light. Let the part OP be in a homo- 

 geneous medium whose index of refraction is /*,, and 



let QR be in a homogeneous medium whose index r , 



of refraction is /n 2 . Between P and Q the ray may o P\ " la & 



pass through any combination of media, singly or 

 doubly refracting. 



Let us consider the characteristic function from a point near to OP in the 

 first medium to a point near to QR in the second. 



