91 



by a name, I shall call them pencils : defining a pencil to be the locus of the rays of a system o 

 the first class, that is, of a system with but one arbitrary constant. 



[23.] Although, as we have seen, an infinite number of pencils may be formed by the rays 

 of a given reflected system, yet there are certain properties common to them all, which render 

 them susceptible of being included in one common analytic expression. For, if we denote by 

 (V) the characteristic function [20.] of the given reflected system, so that 



dV dV dV 



(«, /3, y) being the cosines of the angles which the reflected ray passing through any assigned 

 point of space (x, y, z) makes with the axes of coordinates ; we shall have, for all the points of 

 any one ray, the three equations 



dV dV dV 



— ; — = const. — ; — = const. — ; — = const. 

 ax ay dz 



which are equivalent to but two distinct relations, because 



/dVy /dVV" fdV\^ , 



If then we consider the rays of any of the partial systems, produced by establishing an ar- 

 bitrary relation between the rays of the entire reflected system ; the locus of these rays, that 

 is, the pencil of this partial system, will have for its equation 



f = / (f ) 



dy 



f representing an arbitrary function, the form of which depends upon the nature of the partial 

 system. 



[S*.] The form of this function (/) maybe determined, if we know any curve through 

 which the rays of the pencil pass, or any surface which they envelope. For first, the latter of 

 these two questions may be reduced to the former, by determining upon the enveloped surface 

 the locus of the points of contact ; this is done by means of the formula 



= 0. 



which expresses that the rays of the unknown pencil are tangents to the given enveloped surface 

 M = 0. And when we know a curve u =r 0, « = 0, through which all the rays of the pencil 

 pass, we have only to eliminate (x, y, z) between the two equations of this curve, and the two 

 following, 



_ dV _ dV 



dx dy ' 



