96 



*'» j8'> y being the cosines of the angles that the incident ray, measured from the mirror, makes 

 with the axes of coordinates ; and let p, q, r, s, t, be the partial differential coefficients of the 

 mirror, of the first and second orders, so that 



dz rzpdx + qdi/, dp = rdx -{- sdi/, dq = sdx + tdy, 



X, y, z being the coordinates of the mirror. Then, by the first section of this essay we shall 

 have the two equations 



<^ + »' + p iv+v') = 0, ^ + /3' + y^y+y') = 0, 



which gives by differentiation, 



d'V d^V ^ / d^V (PV'\ , fd'V d^V\ 



d^(v+v') . ^(r+r) , rf2(F+.K'). d^(v+v') 



dx.dy ^^ dy.dz ' ^' dxdz ^ ^' dz^ 

 Combining these three equations with the three which result from differentiating the equation 



(dvy fdVy , /dvy 



^dy i 



we shall have the partial differential coefficients, second order, of F, when we know those of V 

 and of z, that is, when we know the incident system and the mirror; it will then remain to substi- 

 tute them in the formulae of the preceding paragraph, in order to find the developable pencils, 

 and the caustic surfaces, in which we may change the partial differential coefficients of F, taken 

 with respect to (a, b, c), to the corresponding coefficients with respect to (x, y, z). 



[30.] Suppose, to give an example of the application of the preceding reasonings, that the inci- 

 dent rays are parallel, and that we take for the axes of {x) and (y), the tangents to the lines of cur- 

 vature on the mirror at the point of incidence, so that the normal at that point shall be vertical ; 

 the partial differential coefficients of the second order, of (F') will vanish, aud we shall have 



x=0, ,y = 0, z = 0, p=0, q = 0, s = 0, 



« + «' = 0, /3.f^' = 0, y = 7' = C0S. 7. 



/ being the angle of incidence ; the formulae for the partial differential coefficients of the second 

 order of ( V) become 



dW „ d^V „ d'^V 



= — 2y. •:r-7-=o, -— - = — 2y<, 



do? dx,dy dif- 



d%V ^ dW ^ cP-V ^Ur-Y^H) 



dx.dz dy.dz ds? y 



