90 



((t.dx + fiuit/ + y.dz) 



an exact differential, independently of any relation between (x, y, t) ; that is, the cosines («, /3, 

 y) of the angles which the ray passing through any assigned point (or, y, z) makes with the axes, 

 must be equal to the partial differential coefiBcients 



dV dV dV 

 dx ' dy ^ dz ' 



of a function of (x, y, z) considered as three independent variables. 



[20. ] The properties of any one rectangular system, as distinguished from another, may all 

 be deduced by analytic reasonings from the form of the function (V) ; and it is the method of 

 making this deduction, together with the calculation of the form of the characteristic Junction 

 (V) for each particular system, that appears to me to be the most complete and simple definition 

 that can be given, of the Application of analysis to optics; so far as regards the systems pro- 

 duced by ordinary reflection and refraction, which, as I shall shew, are all rectangular. And 

 although the systems produced by extraordinary refraction, do not in general enjoy this property ; 

 yet I shall shew that with respect to them, there exists an analogous characteristic function, from 

 which all the circumstances of the system may be deduced : by which means optics acquires, 

 as it seems to me, an uniformity and simplicity in which it has been hitherto deficient. 



V, On the pencils of a Reflected System. 



[21.] When a rectangular system of rays, that is a system the rays of which are cut per- 

 pendicularly by a series of surfaces, is reflected at a mirror, we have seen that the reflected sys- 

 tem is also rectangular; the rays being cut perpendicularly by the surfaces of constant action, 

 (III.) ; and that therefore the cosines of the angles which a reflected ray makes with the axes, are 

 equal to the partial differential coefficients of a certain function ( V) which \ have called the cha- 

 racteristic of the system, because all the properties of the system may be deduced from it. It is this 

 deduction which we now proceed to ; and before we occupy ourselve^ with the entire reflected 

 system, we are going to investigate some of the properties of the varioiis partial systems that can 

 be formed, by establishing any assumed relation between the rays, that is by considering only 

 those which are reflected from any assumed curve upon the mirror. 



[22. ~j A partial system of this kind, is a system of the first class; that is, the position of a 

 ray in such a system, depends only on one arbitrary element; for example, on any one coordi- 

 nate of the assumed curve upon the mirror. And if we eliminate this one element, between the 

 two equations of the ray, we shall obtain the equation of a surface, which is the locus of the rays 

 of the partial system that we are considering. The form of this surface depends on the arbitrary 

 curve upon the mirror, from which the rays of the paitial system proceed ; so that according 

 to the infinite variety of curves which we can trace upon the given mirror, we shall have an infi- 

 nite number of surfaces composed by the rays of a given reflected system. And since these sur- 

 'aces possess many important properties, which render it expedient that we should denote them 



