330 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
(58.) may be transformed into 
C dft sinix 
^ ^ COS/A ~^^COS^/x’ 
the well-known formula for the rectification of a plane parabola. When, on the other 
band, the sphere becomes indefinitely small compared with the parabola, 7 approxi- 
mates to a right angle, and (58.) becomes 
5=|!//-[-tan"'(tanj77) = 2|M/, 
as it should be, since 2 ;o(; is the angle which the radius vector p makes with the axis. 
We shall find the notice of these extreme cases useful. 
XXI. Although we have called this curve the spherical parabola, as indicating its 
mode of generation, it is in fact a closed curve, like all other curves which are the 
intersections of cones of the second degree with concentric spheres. It is a spherical 
ellipse, and we shall now proceed to determine its principal arcs. 
Let ADG be a parabola, F its focus, O being 
the centre of the sphere which touches the plane 
of the parabola at F, and being also the vertex of 
the obtuse-angled cone, of which the parabola 
ADG is a section parallel to the side of the cone 
OB. Let the angle AOF or the arc Fa be 7 , a 
and (3 being the principal seraiangles of the cone, 
2a=^+7=AOB, 
whence tan^o; = 
1 — sin 7 
To determine the angle (3, or the arc C6. Bisect 
the vertical angle AOB of the cone by the line 
OD, and draw DG an ordinate of the parabola. 
OF 
isosceles triangle, AD=AO=^^; and 
Fig. 6. 
Then tan^(3 = ^Q^^ . As AOD is an 
OF 
OD = ^= 
OF 
sm« 
We have also, as DG is an ordinate of the parabola, 
DG’= 4 AFxAD= 40 F.tanyX —= 421 !^:. 
' C0S7 cos'^7 
Hence substituting, tan^/3=|^;^^^- 
We may therefore announce the following important theorem : — 
7 he spherical ellipse, whose principal arcs are given by the equations 
I -)- siny 
2 siny 
siny’ 
{69.) 
