304 
ME. A. CAYLEY’S MEMOIE UPOX CAUSTICS. 
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 hne. 
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 cirn'e. 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 «, 5, c, d be the values of x corresponding to the points of inter- 
section with the axis, then the equation of the curve is 
y*-\-y'^\2x‘^—{a-{-h-\-c-\-d)x—\{a^-\-lf-\-(?-\-d‘^—2ab~‘lac—2ad—2lc—‘lhd—1cd)~\ 
-\-{x—a){x—h){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 
d-\-d for h, c, d, we may therefore -without loss of generality' put 
a-\-h-\-c-\-d=-^^ and the equation of the curve then becomes 
y^ -\-y\ 2x^-\-ab-\-ac-\-ad-\-hc-\-hd-\-cd)-{-{x— a){x —h){x—c){x—d)—^, 
where 
The curve is in this case said to be referred to the centre as origin. 
The last-mentioned equation may be written 
{x‘^-\-y'^f-{-{ab-\-ac-\-ad-\-bc-\-hd-\-cd)[ci^-\-y^) 
—{abc-\-abd-{- acd -\-bcd)x-^abcd=.0, 
or 
{x^-^y'^-\-\{ab-\-ac-\-ad-\-bc-\-bd^cd)Y 
~ ( abc -\-abd-\- acd + bcd)x 
^ cdb‘^-\- cdc^ -|- <fd^ -\-b'^c'^-\-b‘^d^-\-(Pdj‘ ^ 
^ -\-2a^bc-\-2a%d-\-2a^cd-\-2b^ac-\-2b^ad-\-‘Idfcd ^ 
-\-2c^ab-\-2(fad-\-2c^bd-\-2d^ab-\-2d^ac-\-2d^bc 
-\-2abcd 
or observing that 
a^bc + a^bd + a^cd -\-b^ac-\- b^ad -|- h^cd 
-j- c^ab -f c^ad -\-c‘^bd-\- d^ab + d^ac + d'^bc 
=abc{a-\-b-\-c)-\-abd{a-{-b-\-d)-^acd{a-\-c-\-d)-\-bcd{b-\-c-\-d) 
=.—Aabcd^ 
the equation becomes 
{3(^-\-y‘^-\-\{ab-\-ac-\-ad-\-bc-\-bd-\-cd)Y 
— {abc-\-abd-\-acd-\-bcd)x 
— a^b'^ -f + cdd^ + b'^c^ -j- b'^d^ -f- (fd ^ — 6 abed) — 0, 
