[ S3 ] 
IV. Researches on the Geometrical Properties of Elliptic Integrals. 
By the Rev. James Booth, LL.D ., F.R.S. fyc. 
Received November 17, 1851, — Read January 22, 1852. 
Section XI. — On the Quadrature of the Logarithmic Ellipse and of the Logarithmic 
Hyperbola. 
lxxxiv. In the former part of this paper, printed in the Philosophical Transactions 
for the year 1852, the author has shown that the geometrical types of those integrals, 
named by Legendre and others elliptic functions, are the curves of symmetrical 
intersection of surfaces of the second order. In the progress of those investigations 
he discovered two curves, which he called the Logarithmic Ellipse and the Log- 
arithmic Hyperbola. The properties of these curves have the same analogy to the 
paraboloid of revolution that spherical conics have to a sphere, or which ordinary 
conic sections bear to a plane. To determine the areas of those curves, or rather 
the portions of surface of the paraboloid bounded by them, appeared to the author a 
problem not undeserving of investigation. 
The logarithmic ellipse is defined as the 
curve of intersection of a paraboloid of 
revolution with an elliptic cylinder whose 
axis coincides with that of the paraboloid. 
The logarithmic hyperbola, in like man- 
ner, may be defined as the curve of inter- 
section of a paraboloid of revolution with 
a cylinder whose base is an hyperbola, and 
whose axis coincides with that of the para- 
boloid. 
Through the vertex Z of the paraboloid 
let two parabolas be drawn indefinitely 
near to each other, ZP, ZQ, and let two 
planes indefinitely near to each other at 
right angles to the axis OZ cut the parabolas in the points u, u', v, v'. 
I he little trapezoid uvu'v' is the element of the surface, and if the normal un 
makes the angle g with the axis OZ, d being the elementary angle between the 
planes, uii—k tangd-^, k being the semiparameter of the generating parabola. 
Now uv=ds—k-gg^-^. Hence the elementary trapezoid uvu'v ' = | 
Fig. 27. 
z 
Integrating this expression, area = 
. (436.) 
