362 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Section V. — On the Logarithmic Hyperbola. 
XLV. The Logarithmic hyperbola may be defined as the curve of symmetrical in- 
tersection of a paraboloid of revolution with a right cylinder standing on a plane 
hyperbola as a base. 
Let Oxx^ be a paraboloid of revolu- 
tion, whose vertex is at O, and whose 
axis is OZ. Let ACB be an hyperbola 
in the plane of xy, whose vertex is at A, 
whose asymptots are the lines OX, 
OY, and whose axis is the right line 
GAD. Let the planes ZOX, ZOD, 
ZOY cut the paraboloid in the plane 
parabolas Oo?, Od, Oy, and let cah be 
the curve on the surface of the para- 
boloid whose orthogonal projection on 
the plane oi xy is the plane hyperbola 
ABC. Then ach is the logarithmic 
hyperbola. 
As OX is an asymptot to the hyper- 
bolic arc AB, it is manifest that the 
parabolic arc Ox is a curvilinear 
asymptot to the arc ah of the loga- 
rithmic hyperbola. 
,2 2 
Let 12= 1, and x^-l-i/^=2A’^ (220.) 
be the equations of the hyperbolic cylinder and of the paraboloid of revolution, and 
consequently of the curve in which they intersect. Let T be an arc of this curve, 
then 
T=l 
( 2 - 2 ..) 
x,y, z being functions of a fourth independent variable X. 
„ a!^ cos^X „ sin^X 
Assume 
'a^cos^X — U 
( 222 .) 
It is manifest that these assumptions are compatible with the first of equation (220.), 
and the second of that group gives 
cos^X + sin^X 
cos‘'^X — sin'^X 
= 2kz 
* We migtit, by the help of the imaginary transformation sin0= 'C— 1 tan0', pass at once from the ellipiic 
cylinder to the hyperbolic cylinder. Let tan0'=M, and the resulting equation will be of the form 
(JY o; + ^u- + yu* 
dw V A + Bm- + Cw* + Dm® 
an expression which, on trial, it would be found very difficult to reduce. The difficulty is eluded by making 
the transformation pointed out and adopted in the text. 
