DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 329 
Let a sphere be described touching the plane of the parabola at its focus. The 
spherical curve which is the intersection of the sphere with a cone, whose vertex is at 
its centre, and whose base is the parabola, may be called the spherical parabola. 
To find the polar equation of this curve. 
• ^CJ 
The polar equation of the parabola, the focus being the pole, is being 
the parameter of the parabola. Let y be the angle which g subtends at the centre of 
the sphere, and p the angle subtended by r, then 
2 tany 
tanp = i + cosw' 
Let p be the perpendicular from the focus on a tangent to the parabola, pij the angle 
which this perpendicular makes with the axis of the parabola; Whence in the 
spherical curve, as jo=A:tannr, g=kt3,ny, 
tan®= 
tany 
cosp. ’ 
(56.) 
whence 
sinz« = 
siny 
V 1 — cos^y sin^|W. 
(57*) 
Introduce this expression into the general form for spherical rectification, cr=/sinz:rd|M.+r, 
dz? 
given in (32.), we use the positive sign with r, since 1=^’ 
Now as r, and pj are the sides and an angle of a right- 
angled spherical triangle, since 2pj=u, we get, by Napier’s 
rules, tanr=sin® tan|U;, whence, by substitution, 
. . +tan-r -,-^’”^*:T , 1. . (58.) 
J v^l — cos^ysin^/x L -/! — cos^y sm^/xj '' 
When the sphere becomes indefinitely great the spherical 
parabola approaches in its contour indefinitely near to the 
plane parabola, k being the radius of the sphere, 
siny=tany=|5 
Fig. 5. 
since y in this case is indefinitely small, whence cos^y = 1. In this manner, since s-=k(s, 
* The expression for a perpendicular arc of a great circle let fall from the focus of a spherical ellipse on an 
arc of a great circle a tangent to this curve, is 
sin^tzr— ^ cos'^e cos^fx + (sin-g — sin^e) cos2g+sin£ cose cosix V' sin^'iK — sin^2g sin^/x 
(1 — sin'^2£ sin^/x) 
a being the principal major arc, £ the focal distance, and /x the angle which ■ns makes with «. 
When the curve is the spherical parabola, a, + s=-, a—£=:y, 2s=-—y, and the preceding expression, when 
• . siny 
we introduce these relations, will take the very simple form, sin'Z3'= — > or sinw = l, as we take 
^ V 1 — cos^y sm-f^ 
the sign — or +. See Theory of Elliptic Integrals, p. 31. 
2 u 2 
