75 



XVI. Characteristic Function. 



Tlie systems produced by ordinary reflexion and refraction being all rectangular, the proper- 

 ties of every such system may be deduced from the form of one characteristic function, whose 

 partial differentials of the first order, are proportional to the cosines of the angles that the ray 

 makes with the axes . ■ . . .82. 



XVII, Principal properties of a refracted system. 



The results contained in the preceding part, respecting the pencils of a reflected system, the 

 lines of reflexion, the caustic curves and surfaces ; the osculating focal surfaces, the axes of the 

 system, the principal foci, the images, aberrations, and density ; may all be applied, with suit- 

 able modifications to refracted systems also . . .83, 84', 85, 86. 



XVIII. On the determination of reflecting and refracting surfaces, by their lines of reflexion 



and refraction. 



Analogy to questions in the application of analysis to geometry . . 87. 



Remarks on a question of this kind, which has been treated by Malus ; solution of the same 

 question on the principles of this essay ... . 88, 89. 



Questions of this kind conduct in general to partial differential equations of the second order ; 

 another example, which conducts to a case of the equation of vibrating chords 90. 



The partial differential equation, which expresses the condition for the lines of reflexion or 

 refraction coinciding with one another, resolves itself into two distinct equations ; the surfaces 

 represented by the integral, are the focal reflectors or refractors . , 91. 



XIX. On the determination of reflecting and refracting surfaces, by means of their caustic 



surfaces. 



Object of this section . . . . . 92. 



Remarks on the analogous questions treated of by Monge . . 93. 



Method of reducing to those questions, the problems of the present section . 94. 



Another method of treating the same problems, which conducts to partial differential equations 

 of an order higher by unity ; example, in the case where it is required to find a mirror, which 

 shall have one set of its foci upon a given sphere, the incident rays being parallel ; the complete 

 integral, with two arbitrary functions, represents here the enveloppe of a series of paraboloids, 

 • which have their foci upon the given sphere ; there is also a singular primitive, of the first order, 

 representing the mirrors which have the sphere for one of their caustic surfaces, . 95, 96. 

 - Generalization of the preceding results .... 97. 



