FOCAL LINES OF A REFRACTED PENCIL. 333 



If we now turn the system round the axis of y, through an angle 6 

 from z towards x, we find 



"* 9 y"~ xv cos 9 



(2x cos 6 + z sin 6} z sin 9 yz sin 6\ , . 



~~~ ~~ ' 



2. Now consider two portions of a pencil, close to the origin but in 

 different media, whose indices of refraction are ft, ft respectively, and let the 

 coefficients belonging to these portions be distinguished by corresponding suffixes 

 (, and ,). 



Let the media be separated by the surface whose equation is 



Then, since the characteristic function is continuous at this surface, we 

 have the condition V l F a = when z has the value given in (5). 



Substituting this value of z in the expressions for V l and F 2 , and neglecting 

 terms of the third degree in x and y, we obtain 



B 



cos 0i ~ ^ cos ^ + x (f* 1 sin 0l ~ ^ sn 



ar'cos'g. , y 1 ,xycoa0\ /x 3 cos* 0, , f , xycos0,\ n , . 

 ~2ZT f 2^, 4 2^~/" l "' l> \"pr 22T. ~2CT/ 



That the term in x may vanish, we must have 



& tan 0! = ft sun 0, ................................ (7), 



the ordinary law of refraction. 



Equating to zero the coefficients of x*, y\ and xy, we find 

 cos'0, cos*0, 







/ 



, sm ^, -4, sui 



/ ,/) ,/j\ / \ 



"TlOOtft OOtft) ..................... (8), 



A v 



p- -S-JT -fl=-i(cot^-cot(?,) ..................... (9), 



6, sin P! 5., sin P, 5 v 



COt 0, COt 0, 



