60 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
When, moreover, the surfaces are concentric and symmetrically placed, the pre- 
ceding- expression may still further be simplified to 
which is the g-eneral form for elliptic integrals. 
We can perceive therefore that the solution of the general problem, to determine the 
length of the curve in which two surfaces of the second order may intersect, investi- 
gated under its most general form, far transcends the present powers of analysis. 
It is only when one of the surfaces becomes a plane, or when they are concentric and 
symmetrically placed, that the problem under these restricted conditions admits of 
a complete solution. 
We may hence also surmise how vast are the discoveries which still remain to be 
explored in the wide regions of the integral calculus. We see how questions which 
arise from the investigation of problems based on the most elementary geometrical 
forms — surfaces of the second order — baffle the utmost powers of a refined analysis, 
with all the aids of modern improvement. It is not a little curious, that nearly all 
the branches of modern analysis, such as plane and spherical trigonometry, the 
doctrine of logarithms and exponentials, with the theory of elliptic integrals, may all 
be derived from the investigation of one geometrical problem, — To determine the 
length of an arc of the intersecting curve of two surfaces of the second order. 
LXXXVIII. In the logarithmic hyperconic sections, we may develope properties 
analogous to those found in the spherical and plane sections, if we substitute para- 
bolic arcs for arcs of great circles in the one, and for right lines in the other. Here 
follow a few of those theorems. 
1. From any point on a parabolic section of the paraboloid let two parabolas be 
drawn touching the logarithmic ellipse or the logarithmic hyperbola, the parabolic 
arcs joining the points of contact will all pass through one point on the surface of the 
paraboloid. 
2. If a hexagon, whose sides are parabolic arcs, be inscribed in a logarithmic 
ellipse or logarithmic hyperbola, the opposite parabolic arcs will meet two by two on 
a parabola. 
3. If a hexagon, whose sides are parabolas, be circumscribed to a logarithmic ellipse, 
the parabolic arcs joining the opposite vertices will pass through a fixed point on the 
surface of the paraboloid. 
4. If through the centre of a logarithmic ellipse or logarithmic hyperbola two 
parabolic arcs are drawn at right angles to each other, meeting the curve in two 
points, and parabolic arcs be drawn touching the curve in these points, they will 
meet on another logarithmic ellipse or logarithmic hyperbola. 
5. If a circle, whose radius is a, be described on the surface of the paraboloid, and 
therefore touching the logarithmic ellipse or the logarithmic hyperbola at the extre- 
mities of its major axis, and from the extremities of any diameter two parabolic arcs 
