Tracing Caustic Cu 



rves. 



263 



advantage of being given p is seen, as by the direction of 

 the curve one can tell whether it is necessary to find any 

 more intermediate values of fP. 



It will be found that if a^~ x , and if one reflexion occurs 



o 



at each point of the circle, there will be four cusps, all 



within the circle, as shown in fig. 3 where a=l*5 and r = 5. 



The table below the diagram gives all the values required. 



Fig-. 3. 







a=l-o, 



r=b. 









9. 



$'. n. 



P'- 



/P. 



p. 



P- 











5 



- 9375 



4-0625 











59° 54' 38" 



15° 2' 41" 



4-8286 



-•5734 



4-2552 



1-2979 



•4466 



90° 



17° 27' 27" 



4-7697 







4-7697 



1-5 







155° 



7° 17' 2" 



4-9597 



3-0090 



7-9687 



•6339 



-5-9571 



180° 



| 5 



3-75 



8-75 











It is advisable in cases of reflexion as in fig. 3 to tabulate 

 the values when -^ = 90°, for the reflected ray PP' in that 

 case marks the boundary of the caustic in one direction. 



When -\^ = 90°, 6 and cf> are easily found, for 



sin<£=- -sin0, and 2<£' = 9O°-0, 



2 sin 2 (£=1 -cos 2(£' = l-sin<9, 



2a 2 sin 2 # + r 2 sin 0—^=0. 



The solution of this quadratic gives sin 6, and sinc£ = 



a . n 

 — -sin V. 



