312 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
sections. It accordingly became necessary that I should give especial attention to 
the nature of those curves. I succeeded in showing that the elliptic integral of the 
first order, which is merely a particular case of the circular form of elliptic integrals 
of the third order, represents a spherical conic section whose principal arcs have a cer- 
tain relation to each other. Besides, 1 was so fortunate as to hit upon the true geo- 
metrical representative of an elliptic integral of the third order and logarithmic form. 
I discovered it to be the curve of intersection of a right elliptic cylinder by a para- 
boloid of revolution having its axis coincident with that of the cylinder. These re- 
searches were published in the early part of the present year*. There still remained, 
without investigation, the case when the parameter is negative and greater than I. 
The geometrical representative of this peculiar form, I announced to be a curve, 
which 1 called the Logarithmic Jiyperhola. In the Theory of Elliptic Integrals, p. 1.59, 
I have said, If a right cylinder standing on a plane hyperbola as a base, be substi- 
tuted for the elliptic cylinder, the curve of intersection may be named the logarithmic 
hyperbola. It will have four infinite branches, whose asymptots will be the infinite 
arcs of two equal plane parabolas. This curve, and not the spherical ellipse, is the 
true analogue of the corauion hyperbola,” No demonstration, however, of these pro- 
perties was given in that treatise. 
The main object of the following paper is to prove, that Elliptic Integrals of every 
order, the parameter taking any value whatever hetiveen positive and negative infinity 
represent the intersections of surfaces of the second order. 
To these curves may be given the appropriate name of Hyperconic sections. 
These surfaces divide themselves into two classes, of which the sphere and the 
paraboloid of revolution are the respective types ; from the one arise the circular 
functions, from the other the logarithmic and exponential. The circular integral of 
the third order is derived from the sphere, while the logarithmic function of the same 
order is founded on the paraboloid of revolution. 
Although in the following pages I have, for the sake of simplicity, derived the 
properties of those curves, or of the integrals which represent them, from the inter- 
sections of these normal surfaces, — -the sphere and the paraboloid, — with certain cylin- 
drical surfaces; yet the intersections so produced may be considered as the inter- 
sections of these normal surfaces with various other surfaces of the second order. 
Let U=0 be the equation of the sphere or paraboloid, and V=0 the equation of 
the cylinder. The simultaneous equations U=0, V=0 give the equations of the 
curve of intersection. Let f be any abstract number whatever; then U+/V=0 is 
the equation of another surface of the second order passing through the curve of in- 
tersection. Let U=0 be the equation of a sphere, for example. Accordingly as we 
assign suitable values to the number/*, we may make the equation UH-yV^O repre- 
sent any central surface of the second oi-der. But we cannot, by any substitution or 
* The Theory of Elliptic Integrals, and the Properties of Surfaces of the Second Order, applied to the in- 
vestigation of the motion of a body round a fixed point. London : G. Bell, 1851. 
