CONTENTS. 



Part First : On Ordinary Systems of Reflected Rays. 

 Part Second : On Ordinary Systems of Refracted Rays. 

 Part Third: On Extraordinary Systems, and Systems of Rays 

 in general. 



PART FIRST. ON ORDINARY SYSTEMS OF REFLECTED RAYS. 

 I. Analytic expressions of the Lavi qf Ordinary Reflexion. 



The sum of the cosines of the angles which an incident and a reflected ray, measured from 

 the mirror, make with any assumed line, is equal to the cosine of the angle which the normal 

 to the mirror makes with the same line, multiplied by twice the cosine of incidence ; this theorem 

 determines immediately the angles which a reflected ray makes with three rectangular axes, 

 when we know the corresponding angles for the incident ray, and the tangent plane to the 

 mirror. . . . . . . . Arts. 1, 2. 



Principle of Least Action ; the sum of the distances of the point of incidence, from any two 

 assumed points, situated on the two rays, is equal to the corresponding sum, for any point, inde- 

 finitely near, upon the mirror, (the distances being counted negative when the assumed points 

 are on the rays produced) : consequences respecting ellipsoid, hyperboloid, paraboloid, and 

 plane mirrors, . . • • • 3, 4, 5, 



II. Theory of Focal Mirrors. 



A Focal Mirror is one which would reflect to a given point the rays of a given system ; dif- 

 ferential equation of such mirrors, ..... 6. 



In order that this equation should be integrable, the incident rays must be perpendiculars to a 

 surface, • • • • • • • 7, 8. 



When this condition is satisfied, the integral expresses, that the whole bent path traversed by 

 the light, in going from the perpendicular surface to the Focal Mirror, and from this to the 

 Focus, is of a constant length, the same for all the rays ... 9. 



The Focal Mirror is the Enveloppe of a certain series of ellipsoids . . 10 



