456 
A SUPPLEMENTARY MEMOIR ON CAUSTICS. 
[359 
then after some simple reductions, we find 
>3 = 16 {(a? 2 + y 2 — to 2 — l) 2 + 6m (x — m)}, 
T— 32 (2 (a? 2 + y 2 — m 2 — l) 3 + 18?» (a? — m) (a? 2 + y 2 — ??? 2 — 1) — 27 (x — m) 2 }, 
and thence 
S 3 — T 2 = 1024 (x-myU, 
where 
17= 4 (¿e 2 +y 2 — m 2 —l) 3 
+ 4m 2 (a? 2 + y 2 — on 2 — 1 ) 2 
+ 36m (a? 2 + y 2 — m 2 — 1) (x — on) 
— 27 (x — m) 2 
+ 32 m 3 (a? — m), 
or, what is the same thing, 
U = 
4 (a? 2 + y-J 
— (8m 2 + 12) (x 2 + y 2 ) 2 
+ (3 Qmx + 4m 4 — 20m 2 + 12) (x 2 + y 2 ) 
— 27a? 2 + (— 4m 2 + 18) mx + m 2 — 4 ; 
so that the equation of the secondary caustic is U = 0, or the secondary caustic is, 
as stated above, a sextic curve. 
5. It is easy to see that the foregoing envelope may be geometrically constructed 
as follows: viz. if from the point Q (coordinates cos 6, sin 6) on the reflecting circle 
we draw QM perpendicular to the line x — m = 0, and then from the point M draw 
MN perpendicular to QT, the tangent at T, and produce MN to a point P such that 
PN = NM, then P is a point of the envelope; and we thence obtain for the 
coordinates {x, y) of a point P of the envelope the values 
x = m — 2 (m — cos 6) cos 2 6, 
y = sin 6 — 2 (m — cos 6) cos 6 sin 6,