89 



then the two quantities ft, y, are connected by the relation 



so that one only remains arbitrary, and the system is of the first class. In general if we consider 

 only those rays which belong to a given cone, having the luminous point for centre, and for 

 equation 



<f denoting any given function, the two quantities ^, », will be connected by the given relation 



and the system will be of the first class. If now we suppose a system of rays thus emanating 

 from a luminous point, to be any number of times modified by reflection or refi-action, it is evi- 

 dent that the class of the system will not be altered ; that is, there will be the same number of 

 arbitrary constants, or elemente of position, in the final system as in the original system : pro- 

 vided that we do not take into account the dispersion of the differently coloured rays. But if we 

 do take this dispersion into account, it will introduce in refracted systems a new element of po- 

 sition depending on the colour of the ray, and thus will raise the system to a class higher by 

 unity. 



[18.] From the preceding remarks, it is evident that optics, considered mathematically, re- 

 lates for the most part, to the properties of systems of rays, of the first and of the second class. 

 In the third part of this essay I shall consider the properties of these two classes in the most ge- 

 neral point of view ; but at present I shall confine myself to such as are more immediately con- 

 nected with catoptrics. And I shall begin by making some remarks upon the general properties 

 of those systems, in which the rays are cut perpendicularly by a series of surfaces ; a system of 

 this kind I shall call a Rectangular System. The properties of such systems are of great im- 

 portance in optics ; for, by what I have already proved, they include all systems of rays which 

 after issuing from a luminous point, or from a perpendicular surface, have been any number of 

 times reflected, by any combination of mirrors ; we shall see also, in the next part, that they in- 

 clude also the systems produced by ordinary refraction. 



[19.3 In any system of the second class, a ray may in general be determined by the condi- 

 tion of passing through an assigned point of space, for this condition furnishes two equations be- 

 tween the coefficient of the ray, which are in general sufficient to determine the two arbitrary 

 elements of position. We may therefore consider the cosines («, |8, y,) of the angles which the 

 ray makes with the axes as functions of the coordinates {x, y, 2) of any point upon the ray ; be- 

 cause, if the latter be given, the former will be determined. And if the system be rectangular, 

 that is, if the rays be cut perpendicularly by any series of surfaces, it may be proved by the rea- 

 sonings in Section II. of this essay, that these functions must be of such a nature as to render 

 the formula 



o2 



