DIOPTRIC INVESTIGATION BY GAUSS 

 substituting in (8) 

 whence 



109 



b f =y Y + -m (hg . X O X) ; 

 /."= Y + m (lk X=OX) ; 



" &Y 



(X ON) 



Now substituting in (4) the equation to the refracted ray 

 becomes 



or by (8) 



" 



First: If X be taken sucli that Z /.- (X-ON)=1, i.e. 

 X = O X - ] "-^=0 E, suppose ; 

 tli en when 



; -0 X'-A + r/ ^~ ] =(.) X'+ ] 7' 7 =O E r , 



suppose, 



; ^ 



y=Y r , or P Mild y> are equally distant from the axis. 

 Also, if Y = 0, >j = ; or if a ray proceed from E, it will after 



refraction pass through E'. Also m = - -. l = w", that is, 



I /' (-X- O-N ) 



the ray will be equally inclined to the axis before and after refrac- 

 tion. 



E and E' are called the ' principal points.' 



du 1 



O E = O X - = O X + 



, d u id 



u' a 



O JS "i // \ 7 



du' 



\ 7 / 



'( auw 



OE / =OX'+ 



' ~ 



u u 



d u 



' 



Secondly : If iti" = 0, or the ray be parallel to the axis after 

 refraction, we have from (8) 



b = - m, and the equation to the incident ray becomes 



K 



y + m = in (./ - ( ) X), or .// = in I x - O N - ^ } ; 



