by Refraction of the Circle. 431 



f, c, fju, we have in each case identically the same secondary 

 caustic, the effect of the substitution being simply to interchange 

 the different forms of the equation, and we have therefore iden- 

 tically the same caustic. By writing 



(ft^/*')=(fc*M) 



= *(& c, ft), 

 &c, 



a, /3, y, 8, € will be functional symbols, such as are treated of in 

 my paper " On the Theory of Groups as depending on the sym- 

 bolic equation (Fssl" and it is easy to verify the equations 



a = /3 2 — Sy — eS =ye 

 /3=a 2 = ey = <yS =8e 



7 =b hot = 6/3= /3S SB 0L6 



S = ea =7/3=017 = /3e 

 6=701= 673 =/3y=a6\ 



Suppose, for example, f = — c, i. £. let the radiant point be 

 in the circumference ; then in the fourth system j? = — c, 



d = , (or, since d is the radius of a circle, this radius may be 



A* 



taken -), fjc'=—l, or the new system is a reflecting system. 



This is one of M. St. Laurent's theorems, viz. 



Theorem. The caustic by refraction of a circle when the radiant 



point is on the circumference, is the caustic by reflexion for the 



same radiant point, and a concentric circle the radius of which 



is the radius of the first circle divided by the index of refraction. 



c 2 

 Again, if f m — c/jl, the fifth system gives £'=77, c'=c, film -»lj 



or the new system is in this case also a reflecting system. This 

 is the other of M. St. Laurent's theorems, viz : — 



Theorem. The caustic by refraction of a circle when the di- 

 stance of the radiant point from the centre is equal to the radius 

 of the circle multiplied by the index of refraction, is the caustic 

 by reflexion of the same circle for a radiant point the image of 

 the first radiant point. 



Of course it is to be understood that the image of a point 

 means a point whose distance from the centre = square of 

 radius -s- distance. 



2 Stone Buildings, Nov. 2, 1853. 



