by Refraction of the Circle. 429 



(i-i)^v+/-e 2 )-r+**}+(*-?) 2 +2/ 2 +^-J) 2 



= (*-f) 2 + 2/ 2 + 2c(/*-i) •(£^fj 5 +?+« a (>*~) J 

 or reducing, 



^{(*- ^) 2 + 2/ 2 }= ( ^(^f) 2 + y 2 + ^~i) 2 . 



1 / c 2 \ 

 Again, multiplying the same equation by —%{ ^~~& )> an d a ^d- 



p. / c 2\2 c 2 



ing to each side Ml — ^) + p(^— ? 2 + «/ 2 ), we have 



p((«-B ! +»*) +*( 1 T>) « / («-S*+?'+|( 1 -p)*! 



or reducing, 



(«_ J) f +/ . {| • ( ._ 6 . +f *-+*(i_*)y. 



Hence, extracting the square roots of each side of the equa- 

 tions thus found, we have the equation of the secondary caustic 

 in either of the forms 



to which are to be joined 



^VPfF^- (-?+tV(*-;?y-^ 



+ (|"T)" /(iK_f)2+2 ' 2=a 



Write successively, 



