130 



XIII. Density. 



[66.] Malus, who first discovered that the rays of a reflected system are in general tangents to 

 two caustic surfaces, has given in his Traite D'Optigue, (published among the Memoires des Sa- 

 vons Etrangers) the following method for investigating the density of the reflected light at any 

 given point of the system. He considers two infinitely near pairs of developable surfaces formed 

 by the rays ; and as he believed himself to have demonstrated that the two surfaces of such a 

 pair are not in general perpendicular to one another, when the rays have been more than once 

 reflected, he concludes that the perpendicular section of the parcel of rays comprised between 

 the four developable surfaces, will be in general shaped as an oblique angled parallelogram, 

 whose area is equal to the product of the two focal distances of the section, multiplied by the 

 sine of the angle formed by the two developable surfaces of each pair. He then compares this 

 area with the area over which the same rays would be diffused, if they had proceeded without 

 interruption to an equal distance from the luminous point ; and he takes the reciprocal ratio of 

 these areas for the measure of the density of the reflected light, compared with that of the direct 

 light. The calculations required in this method are of considerable intricacy ; and the most 

 remarkable result to which they lead, is that along a given ray the density varies inversely as the 

 product of the focal distances, being infinite at the caustic surfaces, and greatest at their inter, 

 section. The same result follows from the theorem which I have pointed out in [43.] respecting 

 small parcels of rays bounded by any thin pencil, of whatever shape; and that theorem furnishes 

 a general method for investigating the density of the reflected light, at points not upon the 

 caustic surfaces, which appears to me simpler than that of Malus, and which for that reason I am 

 going here to explain. 



Suppose then that rays issuing from a luminous point have been any number of times 

 reflected by any combination of mirrors; let us put a to represent the density of the 

 direct light at the distance unity from the luminous point, and let us put (j) to represent the 

 space over which any given small parcel of that light, bounded by any thin cone, is perpendi- 

 cularly diffused at that distance. Then, if we represent by (j) the first side of the polygon, that 

 is, the portion of any given incident ray comprised between the luminous point and the first 

 mirror, the perpendicular section of the incident parcel, at that distance from the luminous 

 point will have its area 2 = j"..?; and the space over which the parcel is diffused upon the 



mirror, has for expression — — — , / being the angle of incidence. Immediately after reflexion, 



the parcel will again have its perpendicular section = g^ i=2 ; and if we represent by F' „ F' j, 

 the two focal lengths of the first mirror, that is, the distances from the point of incidence to the 

 two points where the first reflected ray touches the first pair of caustic surfaces, we shall have 

 by [^S.] the following expression for the perpendicular section of the reflected parcel, at any 

 distance (§') from the first mirror; 



