118 Miss A. Everett on a Projective Theorem 



The equations of the ray incident at P(0, y\ z') are 1 



x _ y—y _ z-z m 

 L M ~ N ' 



the equations of the ray refracted at P(0, y\ z') are 



x _y—y _ z ~~ 



the equations of the normal at P are 



x = y-y 

 -i W 



cz 



(2) 



The projection of the incident ray on £ = is 1 



x _ y-y 

 L 



M ' 



(3) 



J 



the projection of the refracted ray on z = is 



X\ _ y — ?/ 



the projection of the normal on z=0 is 



£_ _ y-y' 



-1 by' ' 



The quantities in the denominators may be taken as 

 actual direction cosines, since N is small by hypothesis, and 

 therefore 



And by (1) 



L 2 + M 2 = L 2 -fM 2 + N 2 =l. 



/a'N'-/*N=-^VL , -^L), 



hence N' is of the first order of small quantities like N 

 and z' . Also y' is small, therefore 



l + b 2 y' 2 =l. 



TJie projections (3) of the incident and refracted rays meet 

 at the point (0, y\ 0), or Q say, which satisfies the equation 

 of the surface, neglecting y' 2 . 



The equations of the normal at Q are 



v _y—y' _ z 



