DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 61 
be drawn to any third point on the circle, if one of these parabolic arcs touches the 
logarithmic ellipse or the logarithmic hyperbola, the other will pass through a fixed 
point on the surface of the paraboloid. 
6. If on the paraboloid we describe a circle whose radius is V a 2 +h 2 , and if from 
the extremities of any diameter of this circle we draw parabolic arcs touching the 
logarithmic ellipse or the logarithmic hyperbola, these tangent parabolic arcs will 
meet on the circle. 
These theorems will suffice. There would be little difficulty in extending the list. 
In fact nearly all the projective properties of right lines and conic sections on a plane 
may be transformed into analogous properties of great circles and spherical conic 
sections on the surface of a sphere, and of parabolic arcs and logarithmic sections on 
the surface of a paraboloid. 
Section XII. — On the Rectification of the Lemniscates. 
LXXXIX. There is a particular class of plane curves, of which the letnniscate of 
Bernoulli is an example, to which the principles established in the foregoing pages 
may be applied with much elegance. 
Definition. — This entire class of curves may be defined by the following property. 
The square of the rectangle under the radii vectores drawn from the foci to any 
point on the curve is equal to a constant, plus or minus the square of the semidia- 
meter multiplied by a constant quantity. 
Let Q, Q' be the foci, and O the centre, g, g, r 
the lines drawn from these points to any point 
on the curve. Let OQ — OQ' — c, and let f be a 
variable constant. 
Then by the definition 
ef,=o‘±fV (406.) 
But g 2 g 2 = (x 2 +7/ 2 ) 2 -(- c 4 -f- 2 c 2 y 2 — 2 c 2 x 2 , 
and r 2 —x 2 -\-y 2 , 
hence (x 2 -\-y 2 ) 2 =(f 2 -\-2c?)3c l -\-(f 2 — 2<?)y 2 (467.) 
This is the general equation of the curve, which assumes different forms, as we 
assign varying values to f and c. Some examples may be given. 
(a.) Let c= 0 , orf—co , the equation is that of a circle. 
(b.) Lety >2 >2c 2 , and make t / 2 +2t ,2 =a 2 , f 2 — 2c 2 — h 2 , (468.) 
the equation will become (x 2 -j-y 2 ) 2 =a 2 x 2 -{- b 2 y 2 (469.) 
This is the equation of a curve which may be called the elliptic lemniscate. It is 
the locus, as is well known, of the intersection of central perpendiculars with tangents 
to an ellipse. 
Fig. 28. 
