32
ON THE FLEXURE OF A SPHERICAL SURFACE.
[641
these is & (1 — cosj^), which, however great k is, must be less than the arc of meridian
A"D", = p; substituting for k the value the condition is —„<p, viz. this
sinp
Fig. 2.
Sin^)
is tan ^p < p, which is true for every value up to p — 90 c
should have
№ (1 — cosjp) 2 + E- (k, p) < p 2 ,
viz. writing as before k = , this is
But, more than this, we
sin^)
E 2
sin^)
, p)<p 2 - tan 2 %p ;
this must be true, although (relating as it does to a form of E for which k is greater
than 1) there might be some difficulty in verifying it.
There is, as in the first case, no limit to the value of AB, viz. this may be
= 360°, the spherical zone being then cut along a meridian, or it may be greater
than 360°; and, moreover, the spherical quadrilateral may extend south of the equator,
but of course so that the limiting south latitude does not extend beyond the foregoing
value sin -1 ^: viz. we may have a zone between the latitudes + sin -1 jjj, which may
be a complete zone from longitude 0° to 360° or to any greater value than 360°.
The result is, that the zone is deformed into a surface of revolution, which in its
general form resembles that obtained by the revolution of a half-circle or half-ellipse
about a line parallel to and beyond its bounding diameter, the bounding half-diameter
being less, and the greatest radius of rotation greater, than the radius of the original