174 



CONCLUSION TO THE FIRST PART. 



The preceding pages contain the execution of the first part of 

 our plan ; being an attempt to establish general principles respect- 

 ing the systems of rays produced by the ordinary reflexion of light, 

 at any mirror or combination of mirrors, shaped and placed in any 

 manner whatsoever ; and to shew that the mathematical properties 

 of such a system may all be deduced by analytic methods from 

 the form of one characteristic function : as, in the application 

 of analysis to geometry, the properties of a plane curve, or of 

 a curve surface, may all be deduced by uniform methods from 

 the form of the function which characterises its equation. It 

 remains to extend these principles to other optical systems ; to 

 shew that in every such system, whether the rays be straight or 

 curved, whether ordinary or extraordinary, there exists a Cha- 

 racteristic Function analogous to that which we have already 

 pointed out for the case of the systems produced by the ordi- 

 nary reflexion of light ; to simplify and generalise the methods 

 that we have given, for calculating from the form of this function 

 all the other properties of the system ; to integrate various equa- 

 tions which present themselves in the determination of mirrors, 

 lenses, and crystals satisfying assigned conditions ; to establish 

 some more general principles in the theory of Systems of Rays, 

 and to terminate with a brief review of our own results, and of 

 the discoveries of former writers. But we have trespassed too long 

 at present on the indulgence of mathematicians, and of the Academy, 

 and must defer to another occasion the completion of this extensive 



design. 



W.R.HAMILTON. 



Observatory, 

 April 1828. 



