87 



riation of the first point or origin of the polygon, and by the principle of least action, the part 

 arising from the variation of the several points of incidence, is nothing ; therefore the variation 

 arising from the last point of the polygon must be nothing also, which is the condition expressed 

 by the equation (K,) and which requires either that this last point should be a fixed focus through 

 which all the rays of the final system pass, or else that its locus should be a surface cutting those 

 rays perpendicularly. 



[13.] We see then that when rays issuing from a luminous point, or from a perpendicular 

 surface, have been any number of times reflected, the rays of the final system are cut perpendi- 

 cularly by that series of surfaces, for which 



2 ({) = const. (L) 



2 (j) representing the sum of the several paths or sides of the polygon. When we come to con- 

 sider the systems produced by ordinary refraction, we shall see that the rays of such a system 

 are cut perpendicularly by a series of surfaces having for equation 



2.(m5) = const. 



2. (j»{) representing the sum of the several paths, multiplied each by the refractive power of the 

 medium in which it lies. In the systems also, produced by atmospheric and by extraordinary 

 refraction, there are analogous surfaces possessing remarkable properties, which render it desir- 

 able that we should agree upon a name by which we may denote them. Since then in mecha- 

 nics the sum obtained by adding the several elements of the path of a particle, multiplied each 

 by the velocity with which it is described, is called the Action of the particle ; and since if light 

 be a material substance its velocity in uncrystallized mediums is proportional to the refractive 

 power, and is not altered by reflection : I shall call the surfaces (L) the surfaces of constant 

 action ; intending only to express a remarkable analogy, and not assuming any hypothesis about 

 the nature or velocity of light. 



[14).] We have hitherto supposed all the sides of the polygon positive, that is, we have sup- 

 posed them all to be actually traversed by the light. This is necessarily the case for all the sides 

 between the first and last ; but if the point to which the last side of the polygon is measured were 

 a focus from which the final rays diverge, or if it were on a perpendicular surface situated behind 

 the last mirror, this last side would then be negative; and in like manner, if the first point, or 

 origin of the polygon, were a focus to which the first incident rays converged, or if it were on a 

 perpendicular surface behind the first mirror, we should have to consider the first side as negative. 

 With these modifications the equation (L) represents all the surfaces that cut the rays perpen- 

 dicularly ; and to mark the analytic distinction between those which cut the rays themselves, and 

 those which only cut the rays produced, we may call the former positive, and the latter negative : 

 the positive surfaces of constant action lying at the front of the mirror, and the negative ones 

 lying at the back of it. 



[15.] It follows from the preceding theorems, that if with each point of the last mirror for 



VOL. XV. o 



