85 



and combining these with the relations 



' dx ' dx ' dx ' 

 dec ds dy 



which resu t from the Imown formula 



«2 + /3» + y« = i, 

 we find that the three quantities 



d^ dy dy da da dfi 



dz dy . rfjp rfz ' dy dx ' 



are proportional to («, /S, y), and therefore that the condition (F) resolves itself into the three 

 equations (G). 



[9.] These conditions (G) admit of a simple geometrical enunciation; they express that the 

 rays of the incident system are cut perpendicularly by a series of surfaces, having for equation 



f(»dx + ^dy Jf ydz) = const. (H) 



Let X, Y, Z, be the point in which an incident ray is crossed by any given surface of this 

 series (H), and let j be its distance from the point of incidence (x, y, z) : we shall have 



X. — X = «f, Y — ^ = /Sj, Z — Z = yj, 



and therefore, 



a,.dx + ^'dy + y-dz = — rfg, 

 because .. 



«.rfX 4- /3.rfy + y.rfZ = 0. 



We may therefore put the differential equation of the mirror (E) under the form 



</j + rf5' = 0, 

 of which the integral 



J + {' = const. (I) 



expresses that the whole bent path traversed by the light in going from the perpendicular surface 

 (H) to the mirror, and from the mirror to the focus, is of a constant length, the same for all the 

 rays. In this interpretation of the integral (1) we have supposed the distances, 5, g', positive ; 

 that is, we have supposed them measured upon the rays themselves ; if on the contrary, they 

 are measured on the rays produced behind the mirror, they are then to be considered as negative. 



