62 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
(c.) Let./ 2 =2c 2 . The equation becomes {x 2 -\-y 2 ) 2 =Ac 2 x 2 , or the equation is that of 
two equal circles in external contact. 
(d.) Let / 2 <2c 2 . The equation becomes 
(x 2 -\-y 2 ) 2 ={2c 2 -\-f 2 )x 2 — (2c 2 — -f 2 )y 2 ; and a 2 >b 2 . 
(e.) Let f 2 — 0. The equation becomes {x 2 +y 2 ) 2 = ( lc 2 (x 2 — y 2 ), or the equation is 
that of the lemniscate of Bernoulli. 
(f.) Let / 2 , passing’ through 0, be taken with a negative sign. The equation in 
this case becomes 
{x 2 -\-y 2 ) 2 =.{2c 2 — f 2 )x 2 — (2c 2 -\-f 2 )y 2 , and b 2 >a 2 . 
In one case only does the equation of the lemniscate in its general form coincide 
with that of Cassini’s ellipse ; namely, when /= 0, and h — c, h 2 being the product of 
the radii vectores from the foci. 
The definition of Cassini’s ellipse being “a curve such that the product of the radii 
vectores drawn from two fixed points — the foci — to a third point on the curve, shall 
be constant and equal to h 2 , ,, its equation will obviously be, 2c being the distance 
between the foci, 
h 4 — c 4 = (x 2 -f- y 2 ) 2 — 2 c 2 (x 2 — y 2 ) , 
when h—c, (x 2 -\-y 2 ) 2 =2c l {x 2 — y 2 '). 
This is the equation of the lemniscate of Bernoulli. 
These elliptic lemniscates may also be defined as the orthogonal projections of the 
curves of symmetrical intersection of a paraboloid of revolution with cones of the 
second degree, having their centres at the vertex of the paraboloid. Let a and (3 be 
the principal semiangles of one of the cones. Its equation is 
cot 2 «..z 2 + cot 2 |3.3/ 2 =£ 2 . 
O k olr . n 
Make tana=— , tan/3=— , and the equation of the cone becomes 
a b 
a 2 x 2 + h 2 y 2 — 4 h 2 z 2 . 
Let the equation of the paraboloid be x 2 -\-y 2 =2hz. 
Eliminating z, the equation of the projection of the curve of intersection will become 
( x 2 -\-y 2 ) 2 — a 2 x 2 + h 2 y 2 . 
XC. When the section is an ellipse, the equation of this curve is, as in (469.), 
(x 2 +/) 2 = a 2 x 2 + h 2 y 2 (470.) 
This equation may be put in the form r 2 =a 2 cos 2 A-j-6 2 sin 2 A, 
r being the radius vector, and X the angle it makes with the axis. Let s be an arc 
of this curve ; 
/ ds\ 2 „ /dr\ 2 /c?s\ 2 a 4 cos 2 A + b 4 sin 2 A 
\d\) 1 5 a 1 cos 2 A + b 2 sin 2 A 
since 
