[124 
124] ON A PROPERTY OF THE CAUSTIC BY REFRACTION OF THE CIRCLE. 119 
3F THE 
rtain cases 
of a circle 
nd a very 
,ne in the 
the more 
isidered as 
refraction, 
(as is well 
h may be 
found two 
and taking 
iant point, 
which is of the form 
the envelope is therefore 
C + Aa + Bß = 0 ; 
C 2 = c 2 (A 2 + B 2 ). 
Hence substituting, we have for the equation of the envelope, i.e. for the secondary 
caustic, 
[fj? (tx 2 + y 2 + c 2 ) - (£ 2 + v 2 + c 2 )} 2 = 4c 2 {{y 2 x -1) 2 + (y 2 y - n) 2 }, 
which may also be written 
{¡A (x 2 + y 2 — c 2 ) — (f 2 + 7] 2 — c 2 )} 2 = 4c 2 y 2 (x — £ + y — ■»? ); 
and this may perhaps be considered as the standard form. 
To show that this equation belongs to a Descartes’ oval, suppose for greater con 
venience 7) = 0, and write 
y 2 (x 2 -f y 2 — c 2 ) — | 2 + c 2 = 2Gy V(# — £) 2 + y 2 ; 
1 . / 1\2 
multiplying this equation by 1 , and adding to each side c 2 (y ) + (os — £) 2 + y 2 , 
№ \ yJ 
we have 
1 ” ^ + y2 ~ ^ ~ ? + ^ +( ' X ~ & + 2/ 2 + c 2 ^ V 
= (x- £) 2 + y 2 + 2c (jl - ^ V(x - Z) 2 +y* + c 2 (y- ; 
+2/j = + ,(,-!)} ; 
or reducing 
again, multiplying the same equation by — [1 — , and adding to each side 
y‘ V 1 ” J 2 ) +f5(*~r + n 
we have 
^2 ( 1 “ |) № (« 2 + f ~ ° 2 ) “ I 2 + C 1 +| 2 (® - £ 2 + y 2 ) +(i - | 
a? — =- + 
= | 2 ( (® ~ I) 2 + y 2 ) + J ( :1 - | 2 ) - |) 2 + y 2 + £ (l - |) 2 ; 
y 2 = jj^(*-£) 2 + y 2 + |(l —|)} • 
or reducing,