359] A SUPPLEMENTARY MEMOIR ON CAUSTICS. 457 
or, what is the same thing, 
x = 2 cos 3 0 — m (2 cos 2 0 — 1), 
y = sin 0 (2 cos 2 0 4- 1) — 2m sin 0 cos 0, 
or, as these equations may also be written, 
x = f cos 6 — m cos 26 + \ cos 30, 
y = f sin 6 — m sin 26 + \ sin 30. 
6. This result may be verified by showing that these values satisfy the equation 
cos 20 + 4 (x — m) cos 0 + 4y sin 0 + 2m 2 — 1 — 2 (¿c 2 + y 2 ) = 0, 
and also the derived equation 
sin 20 + 2 (x - m) sin 0 — 2y cos 0 = 0. 
We in fact have 
x sin 0 — y cos 0 = m sin 0 — \ sin 20, 
x cos 0 + y sin 0 = | — m cos 0 + £ cos 20, 
and thence 
(x — m) sin 6 — y cos 0 = — £ sin 20, 
which is one of the equations to be verified; and also 
(x — m) cos 0 + y sin 0 = f — 2m cos 0 + ^ cos 20. 
We have moreover 
x 2 + y 2 = \ + m 2 — 4m cos 0 + f cos 20 ; 
and, by means of these last equations, the other equation 
cos 20 + 4 {x — m) cos 0 + 4y sin 0 + 2m 2 — 1 — 2 (# 2 + y 1 ') — 0, 
is also verified. 
7. The foregoing values of (x, y) give 
dx = (— § sin 0 + 2m sin 20 — | sin 30) d6, — — sin 20 (3 cos 0 — 2m) dd, 
dy — { | cos 0 — 2m cos 20 + f cos 30) d6, = cos 20 (3 cos 0 — 2m) d6, 
or, what is the same thing, dx : dy = — sin 20 : cos 20. 
Hence taking for a moment (X, F) as the current coordinates of a point in the 
tangent of the envelope, the equation of the tangent of the envelope is 
Xdy — Ydx = xdy — ydx, 
or, substituting for x, y, dx, dy their values, this equation takes the very simple form 
X cos 20 — Y sin 20 — 2 cos 0 + m = 0, 
C. Y. 
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