145] 
A MEMOIR UPON CAUSTICS. 
371 
The equation just obtained should, I think, be taken as the standard form of the 
equation of the Cartesian, and the form of the equation shows that the Cartesian may 
be defined as the locus of a point, such that the fourth power of its tangential 
distance from a given circle is in a constant ratio to its distance from a given line. 
XXXIII. 
The Cartesian is a curve of the fourth order, symmetrical about a certain line 
which it intersects in four arbitrary points, and these points determine the curve. 
Taking the line in question (which may be called the axis) as the axis of x, and a 
line at right angles to it as the axis of y, let a, b, c, d be the values of x corre 
sponding to the points of intersection with the axis, then the equation of the curve is 
y 4 + y 2 [2# 2 — (a + b + c + d) x — | (a 2 -f b 2 + c 2 + d 2 — 2ab — 2ac — 2ad — 2be — 2bd — 2cd)] 
+ (x — a) (x — b)(x — c) (x — d) = 0. 
It is easy to see that the form of the equation is not altered by writing x + 6 for x, 
and a + 6, b+ 6, c + 6, d+ 6 for a, b, c, d, we may therefore without loss of generality 
put a+b + c + d = 0, and the equation of the curve then becomes 
y 4 + y 2 (2x 2 + ab + ac + ad + be+ bd + cd) + (x — a) (x — b) {x — c)(x — d)= 0, 
The last-mentioned equation may be written 
(x 2 + y 2 ) 2 -f (ab + ac + ad + bc + bd + cd) (x 2 + y 2 ) — (abc + abd + acd + bed,) x + abed = 0, 
or 
[x 2 + y 2 + £ (ab +ac + ad + bc + bd + cd)} 2 
— (abc + abd + acd + bed) x 
a 2 b 2 + a 2 c 2 + a 2 d 2 + b 2 c- + b 2 d 2 + c 2 d 2 
+ 2 a 2 bc + 2 a 2 bd + 2 a 2 cd + 2 b 2 ac + 2 b 2 ad + 2 b 2 cd 
_ i < 
4 + 2c 2 ab + 2 c 2 ad + 2 c 2 bd + 2 d 2 ab + 2 d 2 ac + 2 d 2 bc 
+ 2abed 
or observing that 
a 2 bc + a 2 bd + a 2 cd + b 2 ac + b 2 ad + b 2 cd 
+ c 2 ab + c 2 ad + c 2 bd + d?ah + d 2 ac + d?bc 
= abc (a + b + c) + abd (a+b + d) 4- acd (a + c + d) + bed (b + c + d) 
= — 4 abed, 
47—2