316 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
TABLE. 
Case. 
Sign. 
Parameter. 
Generating surface. 
Cylindrical surface. 
Hyperconic section. 
I. 
+ 
p=n=<K 
Sphere 
Elliptic cylinder 
Circular sections of 
elliptic cylinder. 
II. 
+ 
U 
o 
g 
II II 
s a 
F 
Sphere 
Elliptic cylinder 
Spherical parabola. 
III. 
+ 
p — n>0 
Sphere 
Elliptic cylinder 
Spherical ellipse. 
IV. 
+ 
II 
s 
II 
o 
Plane 
Elliptic cylinder 
Plane ellipse. 
V. 
— 
p=.m—\— V 1 — 
or m=n. 
Paraboloid indefinitely 
attenuated. 
Circular cylinder 
Circular logarithmic 
ellipse. k 
VI. 
— 
p=zm, or ... 
Paraboloid 
Elliptic cylinder 
Logarithmic ellipse. 
VII. 
— 
p—m — 
Plane 
Elliptic cylinder 
Plane ellipse. 
VIII. 
— 
p—m — i 
Sphere 
Elliptic cylinder 
Spherical parabola. 
IX. 
— 
p=m<\. 
Sphere 
Elliptic cylinder 
Spherical ellipse. 
X. 
— 
p — 1 — 1 
Plane 
Hyperbolic cylinder ... 
Plane hyperbola. 
XL 
— 
II 
V 
Paraboloid 
Hyperbolic cylinder ... 
Logarithmic hyperbola 
XII. 
— 
p=l—\.\. V \ — t'a 
or m=n. 
Paraboloid 
Hyperbolic cylinder. 
Equiparametral loga- 
rithmic hyperbola. 
XIII. 
— 
pz=l=<X} 
Paraboloid 
Vertical plane 
Parabola. 
Cases I., IV., VII., X., XIII. give the formulae for the rectification of the ordinary 
conic sections ; the generating surface in these cases being a plane. When the 
generating surface is a sphere, we get the spherical hyperconic sections ; when a 
paraboloid, the logarithmic hyperconic sections result. 
Section II. — On the Spherical Ellipse. 
III. A spherical ellipse may be defined as the curve of intersection of a cone of 
the second degree with a concentric sphere. 
In the spherical ellipse there are two points analogous to the foci of the plane ellipse, 
such that the sum of the arcs of the great circles drawn from those points to any point 
on the curve is constant. Let a and /3 be the principal semiangles of the cone ; 2a and 
2(5 are therefore the principal arcs of the spherical ellipse. Let two right lines be 
drawn from the vertex of the cone in the plane of the angle 2a, making with the in- 
ternal axe of the cone equal angles g, such that 
cos g= 
cosa 
cos/3 
These lines are usually called focals, or the focal lines of the cone. 
( 2 .) 
The points in 
