116 Professor Hamitton’s Third Supplement 
2 os es SS FS oS e oS os =.) oS 62s a 
2 S78r Vo" Std’ Sy8v Sad \Sode’ Srv * Sadr Sr80) TO Sede" Sadr’ 
os es (ay es SS ss a os (23 . 
or? 60° \drdo" Sadr dadc’ Srdc’ = Sr?_—«\Sado" ‘ 
(X") 
When the point of intersection of the infinitely near initial rays removes to an infinite 
distance, this equation reduces itself to the following, 
es Sg \2 
US Se BSS 
ST sy /8 s eT Sy \? 
-@f--8) 07 -B)-GE-) 
and when in like manner the two infinitely near final rays become parallel it gives the 
following quadratic to determine the two corresponding positions of the point of initial 
intersection, : 
es ss os 2 
Be or ea ieee) 
CoO oD eE 
The caustic surfaces of straight systems, ordinary or extraordinary, were determined 
in the First Supplement: but it seemed useful to resume the subject in a more general 
manner here, and to treat it by the new methods of the present memoir. 
Connexion of the Conditions of Initial and Final Intersection of two Near Paths 
of Light, Polygon or Curved, with the Maxima or Minima of the Time or Action- 
Function V+V,= 3 fvds. Separating Planes, Transition Planes, and Transi- 
tion Points, suggested by these Maxima and Minima. The Separating Planes 
divide the Near Points of less from those of greater Action, and they contain the 
Directions of Osculation or Intersection of the Surfaces for which V and FV, are 
constant ; the Transition Planes toysh the Caustic Pencils, and the Transition 
Points are on the Caustic Curves. Extreme Osculating WVaves, or Action- 
Surfaces: Law of Osculation. Analogous Theorems for Sudden Reflexion or 
Refraction. 
24. The conditions of initial and final intersection of two near luminous paths, 
have a remarkable connexion with the maxima and minima of the integral in the law 
of least action, that is, with those of the characteristic function 74, or rather with 
those of the sum of two such integrals or functions, which may be investigated in 
the following manner. 
