PART FIRST. 



ON ORDINARY SYSTEMS OF REFLECTED RAYS. 



SECTION I. 



Analytic expressions of the lata of ordinary reflexion. 



[1.] When a ray of light is reflected at a mirror, we know by experience, that the normal to 

 the mirror, at the point of incidence, bisects the angle between the incident and the reflected 

 rays. If therefore two forces, each equal to unity, were to act at the point of incidence, in the 

 directions of the two rays, their resultant would act in the direction of the normal, and would be 

 equal to twice the cosine of the angle of incidence. If then we denote by (g/) (j'Z) (n/) the angles 

 which the two rays and the normal make respectively with any assumed line (^), and by (7) the 

 angle of incidence, we shall have the following formula, 



cos. (I + cos. {7 sr 2 cos. 7. cos. nl (A) 



which is the analytic representation of the known law of reflexion, and includes the whole theory 

 of catoptrics. 



[2.] It follows from (A) that if we denote by fx, ^y, ^z, ^x, (y, (z, nx, ny, nz, the angles 

 which the two rays and the normal make respectively with three rectangular axes, we shall have 

 the three following equations. 



COS. ^x -J- cos. f'x = 2 cos. 7. cos. «x"\ 



COS. ^y + COS. {'y = 2 cos. 7. cos. ny\ (B) 



cos. {« + COS. g'z =: 2 cos. 7. cos. nz j 



which determine the direction of the reflected ray, when we know that of the incident ray, 

 and the tangent plane to the mirror. 



[3.] Let {x,y, z) be the coordinates of the point of incidence; x-^'ix, y-^-ly, »+Sa, those 

 of a point infinitely near ; if this point be upon the mirror we shall have 



cos. nx. ix -j. cos. ny. Jy + cos. nz, 3a =: 0, 



2^*2 



