k 2 A 4 (bx 4- ay) (£ 2 — v 2 ) 4- 2 £77 (by — ax) — 0 
(10). 
274 ON THE CAUSTIC BY REFLECTION AT A CIRCLE. [42 
or, arranging the terms •in a more convenient order, 
(■bx + ay) (f 2 — rj 2 ) + 2 (by — ax) £77 — k 2 (b + y) £ + k 2 (a + x) y — 0 (2 / ). 
Hence, considering ff, 77 as indeterminate parameters connected by the equation (1), the 
locus of the curve generated by the continued intersections of the lines (2) will be found 
by eliminating f, 77, A from these equations and the system 
f [A 4- 2 (bx + ay)] + 77 [2 (by — ax)] - k 2 (b + y) = 0 (3), 
£ [2 (by — ax)] + 77 [A — 2 (Ac 4- ay)] + & 2 (a + a’) = 0 (4), 
l 
and from these, multiplying by £, 77, adding and reducing by (2), we have 
- I (6 + y) + V (a + x) - A = 0 (5), 
which replaces the equation (2) or (2'). Thus the equations from which £, 77, A are to 
be eliminated are (1), (3), (4), (5). 
From (3), (4), (5), by the elimination of f, 77, we have 
— X {A 2 — 4 (bx 4- ay) 2 } — 4k 2 (by — ax) (a + x)(b + y) 
— k 2 (a + x) 2 [A + 2 (bx + ay)~\ 
— k 2 (b + y) 2 [A — 2 (bx + ay)] 
+ 4X (by — ax) 2 = 0 (6), 
or, reducing, 
— A 3 + A {4 (a 2 + b 2 ) (x 2 + y 2 ) — k 2 [(a 4- x) 2 + (b + y) 2 ]} 
— 2k 2 (bx — ay) (x 2 + y 2 — a 2 — b 2 ) = 0 (7); 
which may be represented by 
— A 3 + XQ — 2R = 0 (7'). 
Again, from the equations (4), (3), transposing the last terms and adding the 
squares, also reducing by (1), 
k 4 [(a + xf + (b + y) 2 ] = k 2 A 2 + 4& 2 (a 2 + b 2 ) (x 2 + y : ) 
+ 4X [(I 2 “ V 2 ) + ay) + 2^77 (by - ax)] (8) ; 
but from the same equations, multiplying by £, 77 and adding, also reducing by (1), 
k 2 A + 2 (bx 4 ay) (f 2 — 77 s ) + 4^77 (by - ax) 4- k 2 [- £ (b 4- y) 4 77 (a 4- x)] = 0 (9), 
or reducing by (5) and dividing by two,