88 



centre, and with a radius equal to any constant quantity, increased or diminished by the sum of 

 the sides of the polygon path, which the light has traversed in arriving at that point, we construct 

 a sphere, the enveloppe of these spheres will be a surface cutting the final rays perpendicularly. 

 It follows also, that when rays, either issuing from a luminous point, or perpendicular to a given 

 surface, have been reflected by any combination of mirrors, it is always possible to find a focal 

 mirror which shall reflect the final rays, so as to make them all pass through any given point ; 

 namely, by choosing it so, that the sum of the sides of the whole polygon path measured to that 

 given focus, and taken with their proper signs, may be equal to any constant quantity. 



IV. Classification of Systems of Rays. 



[16.] Before proceeding any farther in our investigations about reflected systems of rays, it 

 will be useful to make some remarks upon systems of rays in general, and to fix upon a classifi- 

 cation of such systems which may serve to direct our researches. By a Ray, in this Essay, is 

 meant a line along which light is propagated ; and by a System of Rays is meant an infinite 

 number of such lines, connected by any analytic law, or any common property. Thus, for ex. 

 ample, the rays which proceed from a luminous point in a medium of uniform density, compose 

 one system of rays ; the same rays, after being reflected or refi-acted, compose another system. 

 And when we represent a ray analytically by two equations between its three coordinates, the 

 coeflicients of those equations will be connected by one or more relations depending on the na- 

 ture of the system, so that they may be considered as functions of one or more arbitrary quanti- 

 ties. These arbitrary quantities, which enter into the equations of the ray, may be called its 

 Elements of Position, because they serve to particularise its situation in the system to which it 

 belongs. And the number of these arbitrary quantities, or elements of position, is what I shall 

 take for the basis of my classification of systems of rays ; calling a system with one element of 

 position a system of the First Class : a system with two elements of position, a system of the 

 Second Class, and so on. ,> ^af^;, a i, i:„ h^- 



[17-] Thus, if we are considering a system of rays emanating in all directions from a lumi- 

 nous point (a, b, c), the equations of a ray are of the form 



X — a^ fi{z — c) 

 y — b = t (* — c), 



which involve only two arbitrary quantities, or elements of position, namely ^, », the tangents of 

 the angles which the two projections of the ray, on the vertical planes of coordinates, make with 

 the axis of {«) ; a system of this kind is therefore a system of the second class. If among the 

 rays thus emanating in all directions fi-om a luminous point (a, b, c), we consider those only which 

 are contained on a given plane passing through that point, and having for equation 



