113 



also, that the line joining the luminous point to its conjugate focus, (that is, the axis of the os- 

 culating ellipsoid) passes through the centre of the least osculating circle to the mirror ; and 

 since it is also contained in the plane of the greatest osculating circle, it is tangent to one surface 

 of centres of curvature of the mirror. 



[50.] As another application, let us take the case, of parallel rays reflected by a combination 

 of two given mirrors. Let «, /3, y, «', /3', y', be still the cosines of the angles which the last re- 

 flected and last incident ray, measured from the last mirror, make with the axes of coordinates ; 

 «", /3", y") the given cosines of the angles which the first incident ray, measured towards the 

 first mirror, makes with the same axes ; x, y, z, p, q, r, s, f, the coordinates and partial differ- 

 ential coefficients of the last mirror, and x', y, z', y, q', r', s', t', the corresponding quantities of 

 the first. We have then, 



«' + *" + p'.(y' + y")- =0, /3' -f /3" + /.(y' + y") = ; 



and therefore 



we have also 



rf«' -I- pjyf + (y + y"). dpf = 0. 

 dH'+^.d'/ -f (y-f y"). d^=0; 



x!=zx-]r cc'^', y =y + /S'j', z> = z + y^', 



dx' = rfx + d.{»'^), di/ = dyJf rf.(/3'{'), dz' = dz + d.(yf'), 



rfj' = u!.{dxf — dx) + /i'.(dy' - dy) + y'.{dz' — dz), 



^ being the path traversed by the light, in going from the one mirror to the other ; by means of 

 these equations we can find, for the quantities (dei! -]- pdy'), {d/i' -\- qdy'), which enter into (T), 

 expressions which may be shewn to be of the form 



d»' -j- pdyf = Adx + Bdy 

 rf/3' 4- qdy'= Bdx -I- Cdy, 



A, B, C, involving p, q, ■p, q, r', s', t', g' : and to determine the vertices, the axes, and the 

 principal foci, of the last reflected system, we shall have the following equations, 



e.[(y-J.y).r+^] = l+p^ — {cc + ypf 



i-{(Y+'/)-S + B] = pq—(» + yp)(^+yq) ^ ^^z) 



I 



XI. On the images formed by mirrors. 



[51]. It appears from the preceding section, that when rays issuing from a luminous point 

 have been reflected at a given mirror, the two caustic surfaces touched by the reflected rays in- 

 tersect one another in a finite number of isolated points, at which the density of reflected light 



