CONTENTS OF THE SECOND SUPPLEMENT. 



NUMBERS. 



Introduction. 



Statement and Integration of the Partial Differential Equation, which 

 determines the Characteristic Function of Ordinary Systems of Rays, 

 produced by any number of successive Reflexions or Refiactions, .... 1, 2 



Transformation and Development of the Integral, 3 



Verifications of the foregoing Developments, 4 



Case of a Plane System, 5 



Case of a System of Revolution, 6 



Verification of the Approximate Integral for Systems of Revolution, .... 7 



Other method of obtaining the Approximate Integral, 8 



Connexion of the Longitudinal Aberration, in a System of Revolution, 



vi^ith the Development of the Characteristic Function, 9 



Changes of a System of Revolution, produced by Ordinary Refraction,.... 10 



Example ; Spheric Refraction ; Mr. Herschel's Formula for the Aberra- 

 tion of a thin Lens, , 11 



a 2 



