428 Mr. A. Cayley on a Property of the Caustic 



given by M. Gergonne in the same volume, p. 48. A similar 

 method may be employed to demonstrate the more general 

 theorem, that the same caustic by refraction of a circle may be 

 considered as arising from six different systems of a radiant 

 point, circle, and index of refraction. The demonstration is 

 obtained by means of the secondary caustic, which is (as is well 

 known) an oval of Descartes. Such oval has three foci, any one 

 of which may be taken for the radiant point : whichever be 

 selected, there can always be found two corresponding circles 

 and indices of refraction. The demonstration is as follows : — 



Let c be the radius of the refracting circle, fju the index of 

 refraction ; and taking the centre of the circle as origin, let f , rj 

 be the coordinates of the radiant point, the secondary caustic is 

 the envelope of the circle 



where «, j3 are parameters which vary subject to the condition 



The equation of the variable circle may be written 



{/ A 2 (^ + y 2 + c 2 )-(P + 7; 2 + c 2 )}-2(/x 2 a;-f)a-2( / i 2 y--7 7 )/3=0, 



which is of the form 



C + Aa + B/3 = 0; 



the envelope is therefore 



C 2 =c 2 (A 2 + B 2 ). 



Or substituting, we have for the equation of the envelope, t. e, 

 for the secondary caustic, 



{^V+y 2 +^)-(l 2 +^ 2 +c 2 )} 2 =4c 2 {(^--f) 2 +(^~7 7 ) 2 }, 



which may also be written 



{A*V + y*-**)-(P -f7 7 2 -c 2 )} 2 =4cV(^-P+y-^ 2 ); 



and this may perhaps be considered as the standard form. To 

 show that this equation belongs to a Descartes' oval, suppose 

 for greater convenience 7j = 0, and write 



ti*(a? + y*-(?)-P + c' 2 =2cfjL v/(*-?) 2 + 2/ 2 . 

 Multiplying this equation by 1 %> ana * adding to each side 



flfM J +(#"-?) 2 + y 2 , we have 



