120 Miss A. Everett on a Projective Theorem 



to the plane, so that cos = ~Ll -f Mm + N?2, and the sum of 

 the squares of the denominators = sin 2 6. 



In the present case the equation of the first sagittal 

 plane is 



x sin y\r —y cos yfr = 0, 



thus I = sin yjr, m = — cos^Jr, n=0, p = 0, #' = 0, and 



cos = L sin -^ — M cos ^r. (Substituting these values, we 



find for the projection of the ray (L, M, N) on the first 

 sagittal plane 



x — y' . sim/r . cosa/t . y — i/sin 2 -^ _^ —£ ' 



cos t|t(L cos i/r + M sin^fr) sin ^r(L cos ^r-+ M sin i|r) N 



or 

 



— = -r^-r = ( %f^ | (L cos yJr + M sin i/r) + y' sin i/r. . . (4) 



cos y sm y \ JN / r ' * ' J 



The projection cuts the axis of z at the point where 



,=,'_ , T N f g y. t - < '-.gj^£jr = y-N/rii,t, 



(hcosy + Msm^) cos a 



a being the angle between the two incident rays, assumed 

 small of first order. 



Similarly, if ex! be the angle between the two refracted 

 rays, the projection of the refracted ray on the second 

 sagittal plane cuts the axis of z where 



.j Nysinnf/ 



cos a' 



Hence, to the first order, the rays will meet in a point, 

 N, N', y' being all small. If, in addition, i/r is small this 

 will be true to the second order. 



The point (0, 0, z') satisfies the equation of the refracting 

 surface 0= — 2x-\- by 2 + cz 2 , and the equations of the normal 

 at this point are 



x y _ z — z' 

 =1 ~ i0 ~ ~cz T > 



thus it lies in the plane of zx, i. e. the plane through the 

 normal at O perpendicular to the chief incidence plane. 

 The equat ; onsof refraction for the projections are, from (4) 



\j] cosi/r' — /jl cos yjr __ yJ sin y]r' — fi sin yfr 

 -1 ~~ 0~~ 



_ L' cos^/r' + M'sin yfr' L cos-v/r + M sin ty 



cz' 



= jJb' COS <// — /UL COS </>, 



