318 DR. BOOTH ON THE GEOxMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
We may write this equation in the form 
in 
¥ 
1— sin^p cos2\{/. . 
— ■ 2 -- = • o ( 1 — Sima) ■ 
sin^vf/. . „ 
^(l-sm^/3) 
sin 
or reducing, 
cos^4/ sin^ij' 
sin^p ‘ 
sim« 
sin^/3 ' 
( 6 .) 
This is the equation of the spherical ellipse under another form, which may be ob- 
tained independently, by orthogonally projecting the spherical ellipse on the plane of 
the external axes ; or by taking the spherical ellipse as the symmetrical intersection 
of a right elliptic cylinder with the sphere. 
VI. If in the major principal arc 2a of the spherical ellipse, we assume two 
points equidistant from the centre, the distance s heing determined hy the condition 
cos of the arcs of the great circles drawn from these points 
— the foci — to any point on the spherical ellipse is constant, and equal to the principal 
arc '2a. For a proof of this well-known property, the reader is referred to the 
Theory of Elliptical Integrals, p. 12. 
VII. The product of the sines of the perpendicular arcs let fall from the foci of a 
spherical ellipse on the arc of a great circle touching it, is constant. 
Let ■m and sr' be the perpendicular arcs let fall from the foci on the tangent arc of 
a great circle ; we shall have 
sinw sinro-'=sin(a+s) sin(a— s) (7.)* 
VIII. To find an expression for the length of a curve described on the surface of 
a sphere, whose radius is 1. 
Let u and u! be two consecutive points on the curve, ZQ, 
ZQ' the arcs of two great circles passing through them inclined 
to each other at the indefinitely small angle d^. Through 
u let a plane be drawn perpendicular to OZ, and meeting the 
great circle ZQ' in v. 
Then ultimately uvu’ may be taken as a right-angled tri- 
angle, whence uu'‘^‘='uv^-\-lIv^. 
Now uu'—dff, uv— sinp dd/, vUv—Ap, whence 
d<r= [dp^-f sin ^|0 (8.) 
Integrating this expression between the limits and or and 0, accordingly 
as we take p or for the independent variable, we get 

IX. To apply this expression to find the length of an arc of a spherical ellipse. 
In this case it will be found simpler to integrate the differential expression for an 
* Theory of Elliptical Integrals, &c., p. 13. 
Fig. 1. 
