ME. A. CAYLEY’S MEMOIE UPON CAUSTICS. 
283 
XIII. 
The discussion of the preceding equation presents considerable interest. In the first 
place to obtain the rational form write 
«^~2«^(/3H7^)+(i3^-y7=0, 
this gives 
and we have 
j8^= I — — yJ^y^ 
and consequently 
/3»-y‘=(l ■ 
Hence dividing out by the factor the equation becomes 
(1 _p,»)V-2(l+^“-6^y +3|^»(l +|!4«)/_(1 +^‘)/)23;>+(1 - +(1 +^’)/)’'=0 ; 
or reducing and arranging, 
-l-(I2g-“a^+ (6|a."(l 6|M-^+ 6g/^(I 
which is of the form 
A + - %y?Cy ^ = 0 ; 
and the rationalized equation is 
A^+27;oo^By-2I6^^Cy+5VABC^^=0, 
where the values of A, B, C may be written 
A=(^+/){(I-y)V+(l+y)y}--2(I+y)y^-?/^)+I 
B=4^+3y^ 
C=(I+y)yH?/^) + I; 
the caustic is therefore a curve of the I2th order. 
To find where the axis of x meets the curve, we have 
2/=0, A^=0, 
where 
X. e. 
Ao=(I-y)V-2(I+yy+I 
= {(I-g,)V-I}{(I+g,)V-l)}, 
y=0 
/ . 
1^=+:; , a7= + r-T— ? 
[ — 1— ffc — 1+p 
or there are in all four points each of them a point of triple intersection. 
To find where the line oo meets the curve, we have 
oo, A'^=0, 
2 p 2 
