282 
ME. A. CAYLEY’S MEMOIE UPOX CAUSTICS. 
we have 
cot <p(X— 1) 
1 — k^ sin^ 
)=o. 
and the combination of the two equations gives 
Y= 
X= 
^>(1 — 
$ cot (p — 1 
cot <p — sin^ <p 
cot (p — 1 ’ 
and we have therefore 
. V • Fsin3p($cot(p — 1) . 
,y=Ycos(p+Xsin^= (b =^sm^(p 
w=X cos <p— Y sin <p-. 
<I> cot p — 1 
sin^ sin^ p cos p 
Vsmp ^ ^ 
cot p— 1 
i. e. 
x= 
0(1 — sin'^p) — A:® sin^p cos p . 
O cos p— sin p 
or multiplying the numerator and denominator by (1— A:^sin^f)(Ocosp;-j- sinp;), the 
numerator becomes 
(1—^ sin^ cos (p(l — sin^ <p)—k^ sin"* (p cos (p 
+ 0(sin (p(l — F sin^ sin^ ^ cos (p ) } 
=J{^ sin^ <p cos (p{{l—k^ sin^ (p) — sin^ (p(l —T& sin^ (p ) } 
sin^ (P\/l—k^ sin^ pi(l — F sin^ (p) 
=k^ sin^ (p cos^ p-\-k sin^ p(l--k^ sin^ p>)^, 
and the denominator becomes 
k^ sin^ p cos^ p — (1 — k^ sin^ p) sin^ p 
= sin^ p, 
if k'^=l—k^. 
Hence we have for the coordinates of the point of the caustic, 
jk'^x= —k^ cos® p—k{\ —¥• sin^ pf 
[ y=. k^ sin® p ; 
and eliminating p, we obtain for the equation of the caustic, 
k'^x— — ^®{ 1 — k~-y^}'^ — ^{ 1 — ; 
1 
or writing - instead of k, we find 
(1 )^+^(i -[/TYf 
for the equation of the caustic by refraction of the cii'cle, for parallel rays, lire equa- 
tion was first obtained by St. Laurent. 
