^1 



III. Surface! of Constant Action. 



When rays issuing from a luminous point, or from a perpendicular surface, have been any 

 number of times reflected, they are cut perpendicularly by a series of surfaces, possessing this 

 property, that the whole polygon path traversed by the light, in arriving at any one of them, is 

 of a constant length, the same for all the rays . . . .11,12 



Reasons for calling these surfaces, Sttr/ac« o/" Cons^anf adion . . IS 



Distinction of these surfaces into positive and negative . . . 14. 



Each surface of constant action is the enveloppe of a certain series of spheres; if it be itself a 

 sphere, the final rays all pass through the centre of that sphere ; it is always possible to choose 

 the final mirror, so as to satisfy this condition . . . , 15 



IV. Classification of Systems of Rai/s. 



Elements of Position of a ray ; a system in which there is but one such element, is a si/stem 



of the first Class ; a system with two elements of position, is a system of the second class ; the 



principal systems of optics belong to these two classes . . . 16, 1 7- 



A system is rectangular when the rays are perpendiculars to a surface . 1 8 



In such a system, the cosines of the angles that a ray makes with the axes of coordinates, are 



equal to the partial differentials of a certain characteristic Junction, . 19, 20 



V. On the pencils of a Reflected System. 



The rays that are reflected from any assumed curve upon the mirror, compose a partial system 

 of the first class, and have a /ienc?7 for their locus . . • 21,22. 



An infinite number of these pencils may be composed by the rays of a given reflected system ; 

 functional equation of these pencils, ..... 23. 



The arbitrary function in this equation, may be determined by the condition, of passing 

 through a given curve, or enveloping a given surface ; application of these principles to pro- 

 blems of painting and perspective . . . . .24. 



We may also eliminate the arbitrary function, and thus obtain a partial differential equation 

 of the first order, representing all the pencils of the system ... 25 



VI. On the developable pencils, the two foci of a ray, and the caustic curves and surfaces. 



. Each ray of a reflected system has two developable pencils passing through it, and therefore 

 touches two caustic curves, in two corresponding foci, which are contained upon two caustic 

 surfaces . . . . . . . .26 



Equations which determine these several circumstances . . 27, 28, 29 



Examples ....... 30 



Remarks upon the equations of the caustic curves and surfaces . . . 31 



VOL. XV. M 



