98 



- VII. Lines of Reflection on a mirror. 



[32.] We have seen that the rays of a reflected system are in general tangents to two series 

 of caustic curves, and compose two corresponding series of developable pencils ; the intersections 

 of these pencils with the mirror, form two series of remarkable curves upon that surface, which 

 were first discovered by Malus, and which were called by him the Lines of Reflexion. We pro- 

 pose, in the present section to investigate the differential equation of these curves, and some of 

 their principal properties ; and at the same time to make some additional remarks, on the manner 

 of calculating the foci, and the caustic surfaces. - - 



[S3.] To find the differential equation of the curves of reflexion, we may employ the formula 

 of the preceding section, 



d&\( -£1L_ !!-L\d +( —- J1L\ d \- 



' \\ ' dx.dz ' dx.dy) \ ' dyi dy.dz J ' ' )~ 



^ C/ d'^V d'-V\,^^ I dV d'V\ , ) _ 



Hy'"-d^-''-^''+y''-d^z-^-d^)- '"■ I ' (P5 



considering («, /8, y) as given functions of the coordinates of the point of incidence, such that 



, d'V , , d'V , ^ d^V ^ 



da = , ■ . dx + . dy + — — — . dz. 



dx^ dxdy -^ ^ dx.dz 



d'V d^V , ■ d^V ^ 



and deducing the partial differential coefficients of the characteristic function V, either immediately 

 from the form of that function itself, if it be given, or from the equation of the mirror and from the 

 nature of the incident system, according to the method already explained. But in this latter case, 

 that is, when we are only given the incident system and the mirror, it will be simpler to treat the 

 question immediately, by reasonings analogous to those by which the formula (P) was deduced. 



Let, therefore, X, Y, Z, represent the coordinates of a point upon a caustic curve, at a dis- 

 •tance (5) from the mirror ; we shall have 



X=.X-\-»^, y=y-{. /Jj, Z = z-f-y{, 



rf{ = «. d{X—x) + ^ 4 Y—y) -{- y. d[Z—z), 



dX = cc.{»dX-\-^dY+ydZ) 

 aY=^{»dX-\-^dY-\^dZ) 

 dZ = y\»dX-^^d y+yrfZ), 



dx — ct.(ctdx + lidy-\-ydz) -j- ^d» = 

 dy — /3. {adx-\-^dy-{ ydz)-\- {(f/3=: 

 dz — y.{»dx-\^pdy-)f ydz) -J- jrfy ^ ; 



