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