320 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
If we introduce these values into (14.), the transformed equation will become 
tan/3 
tana 
sin^J[ 
d<p 
[1 — sin^^] V \ — sin®>j sin^ip_ 
(16.) 
an elliptic integral of the third order and circular form, since ^ is greater than sin*;;, 
and less than 1. 
This is case IX. in the Table, page 6. 
This is one of the simplest forms to which the rectification of an arc of a spherical 
ellipse can be reduced. The parameter of the elliptic integral is the square of the 
eccentricity of the plane elliptic base, and the modulus is the sine of half the angle 
between the planes of the circular sections of the cone. 
If we write m for e^, i for sin;?, and express the coefficient sin/3 in terms of m 
and i, the expression (16.) may be transformed into 
m )^^^J[[l-msin2<p] 
It is easily shown that the coefficient^^ sinjS of the elliptic integral in (16.) or its 
equal is the square root of the criterion of sphericity, 
>!=(l-m)(l-y. 
For if we substitute in this expression for i, its value given in (1.) m—n-\-7nn=i-, we 
shall find 
js/“« (IS-) 
As\/;t is manifestly real, the elliptic integral is of the circular form. 
X. We may, by the method of rectangular coordinates, derive an expression for 
the arc of a spherical ellipse. 
In this case we shall consider the spherical ellipse as the curve of intersection of a 
right elliptic cylinder by a sphere having its centre on the axis of the cylinder. 
2 2 
Let ^“1-^=1, -\-y ^ . . . (19.) 
be the equations of the cylinder and sphere, ABCD and 
FGCD, then d<r being the element of an arc on the 
surface of a sphere whose radius is 1, kds will be the 
element of the corresponding arc on the surface of the 
sphere whose radius is k. 
H-ce 
X, y and being functions of the independent variable X. 
