Tracing Caustic Curves. 



261 



In the case of a plane wave-front (f> must be taken as the 

 variable. 



In the general case when 6 is given, sin (/> = gin 6. 



The formulae for all cases can then be tabulated as below. 



Table I. 

 p — r sin $', u = r cos ft ; fJ > = n+p'. 



Refraction. 



Reflexion. 



P' 



■ ^=9+2<p'. 



— ar 



/*/• sec -\-fia sec <p~ a sec (p 



sec 9 -\- 2a sec i 



_ r /i¥> sec 3 e +;Lt 3 a 3 S ec 3 0-a 3 sec 3 a' ^ _ [ r J sec j 9 + 2a J sec J f) 

 P— ^| ~ (/^rsece+^asec^ — a seed') 3 J* P ~ p \ (?'sec 0+2asec ^') 3 J ' 



Plane Wave-front. 



r 



p = , . 



[isecQ — sec 



f /* 3 sec 3 <p — sec 3 0' 

 I (ft sec (p — sec <j>') 3 



Plane Wave-front. 



;//=20' 



2 sec 



= _fra, ,\/P=|w. 



f 2 sec 3 0' 1 



Some hesitation may be felt in accepting these results 

 without further examination ; but all doubts will be removed 

 on showing that /P as here obtained is identical with v x as 

 given in the usual formula : — 



fl COS 2 ft COS 2 <£ _ fl COS ft — cos </> 



Vi it r 



fjuur cos 2 (/>' 

 1 u(/ii cos cj)' — cos <f>) + r cos 2 $ ' 

 On referring to fig. 1, SP or u is seen to be r cos cj) + a cos (9. 



/P = ?'COS$' + 



fir sec + /ma sec <£ — <z sec 0' 

 fir 2 cos $' sec + //,ar cos (£' sec cf> 

 fir sec 6 -\-fjba sec <£ — a sec <// 



fir 2 cos 2 $' cos $ + //-a? 1 cos 2 $' cos 

 yitr cos </> cos ft + /ia cos cos ft — a cos # cos (j> 

 fiur cos 2 <£' 



Z*//, cos ft + ?' cos 2 $ — w cos 



