10 
PEOFESSOE CAYLEY’S SUPPLEMENTAEY MEMOIE ON CAUSTICS. 
and, by means of these last equations, the other equation 
cos 25+4(#— m) cos $+4# sin 5+2m 2 — 1 — 2(^ 2 +^ 2 ) = 0, 
is also verified. 
7. The foregoing values of (x, y) give 
dx =(— f sin 3+ 2m sin 25— f sin 35)^5= —sin 25(3 cos 5— 2 m)d&, 
dy={ f cos 5— 2w cos 25+| cos 35)<Z5= cos 25(3 cos 5— 2m)dQ, 
or, what is the same thing, 
dx : dy = — sin 25 : cos 25. 
Hence taking for a moment (X, Y) as the current coordinates of a point in the tangent 
of the envelope, the equation of the tangent of the envelope is 
~Kdy — Y dx— xdy — ydx , 
or, substituting for x, y, dx , dy their values, this equation takes the very simple form 
X cos 25— Y sin 25—2 cos 5-j-m=0, 
or writing (x, y) in place of (X, Y), that is taking now (x, y) as the current coordinates 
of a point in the tangent, the equation of the tangent is 
x cos 25 — y sin 25 — 2 cos 5-j-wi=0 ; 
whence observing that this equation may be expressed as a rational equation of the 
fourth order in terms of the parameter tan-^5 (or cos5-+-\/ —1 sin 5), it appears that 
the class of the secondary caustic is=4. 
8, The secondary caustic may be considered as the envelope of the tangent, and the 
equation be obtained in this manner. Comparing with the general equation 
A cos 25+D sin 25+C cos 5+D sin 5+E=0, 
we have 
A=*, 
» =-y. 
C = — 2, 
D=0, 
and thence 
giving 
if for a moment 
E —m, 
S=4{3(# 2 +y 2 )+m 2 — 3}, 
T=4{18m(^ 2 +^ 2 )— 27#— 2m 3 +9m}, 
S 3 — T 2 =16V, 
V=4{3(^ 2 +^ 2 )+m 2 — 3} 3 
— {18m(# 2 +y 2 )— 27#— 2m 3 +9m} 2 . 
The equation of the curve is thus obtained in the form V = 0 ; this should of course 
be equivalent to the before-mentioned equation U = 0 ; and by developing Y, and com- 
