322 DR. BOOTH ON THE GEOMETRICAL PROPERTIES OF ELLIPTIC INTEGRALS. 
Now«^=Psina, sin^/S, — cos^a, cos^/3, and 0 :^+ 3 /^ =A'^ cos’/o. 
Reducing, we get 
2 sin^a cos^\I/ + siri^/3 sm^4' 
COS p cos^4' + tan^/3 
Comparing this expression with (13.), we see that 
= (30.) 
XI. In. the foregoing expressions (17*) and (28.) for the rectification of an arc of a 
spherical ellipse, the elliptic integrals are of the third order and circular form, with 
negative parameters. We shall now proceed to show that the same arc may be ex- 
pressed by an elliptic integral of the third order and circular form, having a positive 
parameter. 
It is shown in most elementary treatises on the integral calculus, in its applica- 
tion to the rectification of plane curves, that if p the perpendicular let fall from a fixed 
point as pole on a tangent to the curve, makes the angle X with a fixed right line 
drawn through the pole, t being the intercept of the tangent between the point of 
contact and the foot of the perpendicular, we shall have 
and<=-^ 
(31.; 
The signs of s to be taken as the curve is concave or convex to the pole. 
XII. To investigate an analogous formula for the rectification of a spherical curve, 
the intersection of a cone of any order with a concentric sphere. 
Let a point Z be assumed on the surface of the sphere 
as pole, and through this point a tangent plane ZAQB, 
or ( 0 ), to the sphere being drawn, the cone whose ver- 
tex is at O, the centre of the sphere, and which passes 
through the given spherical curve, will cut this tangent 
plane (0) in a plane curve AQB, whose rectification may 
be effected, when possible, by (31.). Now a tangent 
plane OQP, or (T), may be conceived as drawn touching 
the cone, and cutting the tangent plane ( 0 ) in a right 
line QP or t, which will be a tangent to the plane curve 
in (0). It will also cut the sphere in an arc of a great 
circle (;tro) which will touch the spherical curve in ;c. Let 
the distance QO of the point of contact of the line t 
with the plane curve from the centre of the sphere be R. Through the centre of the 
sphere let a plane OZP, or (11), be drawn at right angles to the straight line t. Now 
this plane, as it is perpendicular to t, must be perpendicular to the planes (0) and (T) 
which pass through t. As the plane (IT) is perpendicular to the plane (0), it must 
pass through (Z) the point of contact of this plane with the sphere, and cut the plane 
