131 



2' as ^■<g'~-^'')^g'~ ^^"> , in which 2 = «».*. (L,'") 



Now let 4' be the second side of the polygon, that is the path run over by the light in going 

 from the first mirror to the second, and let (F,", fa") be the two focal lengths of the second 

 mirror ; we shall have, in a similar manner, for the area of the perpendicular section of the parcel, 

 after the second reflexion, at a distance {" from the second mirror, 



y' lo" — F "\( o" F "\ 



■" — if II p n ' > ' 



and so on, for any number of reflexions. Having thus got the space over which the reflected 

 rays are perpendicularly diffused, the density is obtained by this formula 



^(«) _ fi:^ (N'") 



for instance, if the rays have been but once reflected, then the density is 



^- — -7* («'-f.')(5'-^v)' ^": 



a formula which agrees with that of Malus ; after two reflections, the density is 



^ - E" - (j"_f /')(?" — i-^") ^^ 



a' being the density immediately before the second reflexion: a formula which is different from 

 that of Malus, and which appears to me to be simpler. 



{[67.] The two preceding methods, namely, that of Malus, and that of the preceding para- 

 graph, fail when the density is to be measured at the caustic surfaces ; for they only shew that 

 the density at those surfaces is infinitely greater than at other points of the system, without 

 shewing by what law the density varies in passing from one point of a caustic surface to another. 

 This question, which has not been treated by Malus, appears to me too important to be passed 

 over, although the discussion of it is more difficult than the investigation of the density at ordi- 

 nary points of the system. 



Let us then, as a first approximation, resume the formulae of [60.] 



x', y' being the coordinates of the point in which a near ray crosses the plane of aberration, 

 that is, a plane perpendicular to the given ray, passing through the focus of that ray; «„ ,3,, small 

 but finite quantities, namely, the cosines of the angles which the near ray makes with the axes of 

 {x!) and (y), the former of which axes is a tangent and the other a normal to the caustic sur- 

 face ; A, B, C, coefficients depending on the curvature of that surface, and on the interval (i) 

 between the two foci of the ray. To find by these equations the space over which the rays, 



