PROFESSOR CAYLEY’S SUPPLEMENTARY MEMOIR ON CAUSTICS. 9 
or, what is the same thing, 
U= 4 (x 2 +y 2 ) 3 
-(8m 2 +12) (x 2 +y 2 ) 2 
-j-(36m#+4m 4 — 20m 2 +12) (x 2 -\-y 2 ) 
— 27# 2 -f-(— 4m 2 + 18)m.r+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 3, sin 3) 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 3) cos 2 3, 
y= sin 3 — 2(m — cos 3) cos 6 sin 3, 
or, what is the same thing, 
x=2 cos 3 3— m(2 cos 2 3— J), 
y— sin 3(2 cos 2 3 + 1)— 2m sin 3 cos 3, 
or, as these equations may also be written, 
x—\ cos 3— m cos 23-f ^ cos 33, 
y— f sin 3 — m sin 23+-| sin 33. 
6. This result may be verified by showing that these values satisfy the equation 
cos 23 + 4 {x—m) cos 3 + iy sin 3 + 2m 2 — 1 — 2(# 2 +y 2 ) = 0, 
and also the derived equation 
sin 23+2 (x—m) sin 3—2 y cos 3=0. 
We in fact have 
x sin 3— y cos 3= msin 3— sin 23, 
x cos 3+y sin 3=f —m cos 3+^ cos 23, 
and thence 
(x—m) sin 3— y cos 3= sin 23, 
which is one of the equations to be verified ; and also 
(x—m) cos 3+^ sin 3=f — 2m cos 3+^ cos 23. 
We have moreover 
x 2 - \-y 2 = -f + m 2 — 4m cos 3 + f cos 23 ; 
mdccclxvii. c 
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