103 



mirrors, possessing the property of reflecting the rays to any given point (X, Y, Z), and having 

 for their differential equation, 



Jf = ctfdx + /3'rfy+ -/dz = d V, 



V'h eing the characteristic function of the incident system, and 5 the distance from the point of 

 incidence (x, y, 2) to the point (X, Y, Z), the focus of the focal mirror. The condition of touch- 

 ing the given mirror at a given point, furnishes two equations of the form 



which express that the focus (X, Y, Z) is somewhere on the given reflected ray ; and the condi- 

 tion of osculating in a given direction furnishes the equation 



[r'—r). dx" -f 2(s' — s) dx.dy -f {I' — t). dy* - 0, 

 (r. s, i) being given, but (/, s', f) depending on the unknown focal distance (5) ; and if we wish 

 to make this distance a maximum or a minimum, we are to satisfy the two conditions 



{r'—r). dx + (s' —s). dy = 0, {s' — s). dx + {f — t). dy = 0. 

 which may be thus written 



dpf = dp, dq' == dq, 

 p, q', being the partial differentials, first order, of the focal mirror, and/), ^, those of the given 

 mirror. Now the general equation of focal mirrors, d^=^dV' = cc'dx -J- /i'dy -f- y'dt, gives 



X-X +yiz -Z)= «.{«' + y'p), 



y - y + 9'-(2 -Z) = <i¥ + Vq'), 



and therefore 



dx + p'dz — («' -t- y'p') d^ = ^.{d»' + pdy') + (Z _ 2 -|-yj) rfy 

 dy + c[dz — {0 + ■/?') di = ^.{dH' + q'dv') + (Z — Z+ y'O dq' ; 



if then we put p' = p, 9' =7, in order to express that the focal mirror touches the given mirror, 

 we shall have, to determine dp', dpf, two equations which may be thus written, 



{•{(y+y) dp' -f- dx'+pdy'l = dx +pdz — (« + yp){adx-{-/idy+ydz) 1 



/■ (Z) 



«-{(yH-y') dq' -\-d^' + qdy'] =dy + qdz — (fi + yq)(»dx+^dy + ydz)y 



and if in these equations (Z) we change (dpf, dq') to {dp, dq) in order to find the greatest and 

 least osculating focal mirrors, they become the formulae (T) of the preceding section. Hence it 

 follows, that in general, the foci of the greatest and least osculating mirrors, are the points in 

 which the reflected ray touches the two caustic surfaces ; and that the directions of the lines of 

 reflexion, are the directions of osculation corresponding. 



[39.] The equations (Z) determine not only the maximum and minimum values of the oscu- 

 lating focal distance (^), but also the law by which that distance varies for intermediate direc- 

 tions of osculation. To find this law, we are to employ the formula, 



VOL. XV. a 



