286 
ME. A. CAYLEY’S IklEMOIE EPOY CAESTICS. 
refraction [Jj, is the same curve as the caustic hy refraction for parallel rays of a con- 
centric circle radius index of refraction -• 
ft ft 
XV. 
We may consequently in tracing the caustic confine our attention to the case in which 
Q 
the index of refraction is greater than unity. The circle radius - will in this case be 
within the refracting circle, and it is easy to see that if from the extremity of the diameter 
of the refracting circle perpendicular to the direction of the incident rays, tangents are 
drawn to the circle radius the points of contact are the points of triple intersection of 
the caustic with the last-mentioned circle, and these points of intersection being, as 
already observed, cusps, the tangents in question are the tangents to the caustic at these 
cusps. The points of intersection with the axis of x are also cusps of the caustic, the 
tangents at these cusps coinciding with the axis of x : two of the last-mentioned cusps. 
L C 
\\z. those whose distances from the centre are H he within the circle radius the 
— ft + 1 ft’ 
other two of the same four cusps, viz. those whose distances from the centre are 
Q 
lie without the circle radius - ; the last-mentioned two cusps lie without the refi’acting 
circle, when jM/<2, upon this circle when [z=2, and within it, and therefore between the 
two circles when |M->2. The caustic is therefore of the forms in the annexed figimes 
3, 4, 5, in each of which the outer circle is the refracting circle, and |t/; is >1, but the 
Fig. 3, Fig. 4. Fig. 5. 
three figures correspond respectively to the cases [/j< 2, (jb=2 and f/j>2. The same 
three figures will represent the different forms of the caustic when the inner cfrcle is 
the refracting circle and (Ji> is <1, the three figures then respectively corresponding to 
the cases and 
XVI. 
To find the tangential equation, I retain k instead of its value -•> the equation of the 
refracted ray then is 
x{k cos ■— \/l — sin^ <p) -\-y(k sin <p cot (p \/ 1 — siiE <p) — ^ = 0, 
