ME. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
311 
and the values of the coefficients are 
A=1 
B=0 
D=4|«/^3/ 
E = ~2^V+^^)-2^^+l. 
Substituting these values in the equation 
{12(A^+B")-3(C^+D^)+4E^}"-{27A(C"-D")+54BCD 
-(72(A^+B^)+9(C^+D^))E+8E^}^=0, 
the equation of the envelope is found to be 
_(6^^+3(4^+6^«)(a’^+2/^)-27^V-/) _ 
+ 4/A®(a4+?/®)® 
which is readily seen to be only of the 8th order. But to simplify the result, write first 
and 2^— 1— 1) in the place of and x^—y^ respectively, 
the equation becomes 
4{(l-^7-^^(l-^^X^+y^-l)+^X^^+3/^-l)^>^ 
■ 2(l-f/,7 '> 
Write for a moment — l)=f? the equation becomes 
%^-??+??-(22^-32^f-3^^+2^^-27/AVX=0; 
or developing, 
+54(2^3 - 3^^^- 3^^^+2^*V.V- 729(ry,V= 0, 
and reducing and dividing out by 27, this gives 
2y(f-if+2(f+2)(2j-2)(j-25>V-27A‘=0, 
whence replacing §', g> by their values, the required equation is 
(1 l)=(f.V+2^)-l)’ 
which is the equation of an orthogonal trajectory of the refracted rays. 
