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42] ON THE CAUSTIC BY REFLECTION AT A CIRCLE. 
2 75 
Using this to reduce (8), 
¥ [(a + xf + (b + yf] = 4 (a 2 + b 2 ) (:x 2 + y")-h 3A 2 (11), 
or, from the value of P, 
- 3A 2 + Q = 0 
(12), 
which singularly enough is the derived equation of (7') with respect to A: so that 
the equation of the curve is obtained by expressing that two of the roots of the 
equation (7') are equal. Multiplying (12) by A and reducing by (7(), 
-\Q + SR = 0, 
or, combining this with (12), 
.27 R 2 -Q s = 0; 
whence, replacing R, Q by their values, we find 
27¥ (bx — ay)- (x 2 + y- — a- — b-J 2 — {4 (a 2 + b 2 ) {x- + y 2 ) — ¥ [(a + xf + (b + yf~\| 3 = 0, 
the equation of M. de St-Laurent. 
35—2 
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