IHE. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
309 
(where yi are current coordinates), then 
X:Y;Z=-?/(^+y-«) 
: 4A^, 
equations which give 
Z^Yx =Y(mZ^-AX^) 
-T^Yy =Z^+X(mZ^-AX^) 
Z^Y^(.r"+?/"-a)= 4AZ®XY^ 
whence eliminating, we have 
{Z^+X(mZ^-AX^)}^+YVwZ’--AX7-Z^Y^(«Z + 4.AX)=0, 
w'here if, as before, c denotes the radius of the refracting circle, a the distance of the 
radiant point from the centre, and Uj the index of refraction, we have 
The above equation is the condition in order that the line Xii‘-f Y;^ + Z = 0 may be a 
normal to the secondary caustic {ci^-\-y’^ — af-\-l%Y[x—'in)=0^ or it is the tangential 
equation of the caustic, which is therefore a curve of the class 6 only. The equation 
may be written in the more convenient form 
Z«+2Z^X(?nZ^-.AK^)+(X^+Y^)(mZ^-AX^)--^-Z^Y^(aZ + 4AX)=:0. 
XXXVIII. 
To compare the last result with that previously obtained for the caustic by reflexion, I 
write |M<= — I, and putting also c=l and Z=« (for the equation of the reflected ray was 
assumed to be Yxc-\-Yy-\-a=^), we have 
c£=a"+2, A=i«, m=^(I+2«"), 
and the equation becomes, after a slight reduction, 
4a^+4a^X(2«^+l-X^)+(X^+Y^)(2«^+l-X^)^-4«^Y^V+2+2X)=:0, 
which may be wTitten 
(2ffl^+X(2«^+l-X^))^+Y^(-4a'^+l-8a^X-2(2«^-fl)X‘^+X^) = 0; 
this divides out by the factor (X-f 1)^, and the equation then becomes, 
(X^ - X - + Y2((X - 1 - 4«-) = 0, 
which agrees with the result before obtained. 
