[641 
641] 
ON THE FLEXURE OF A SPHERICAL SURFACE. 
31 
RFACE. 
8—90.] 
ideas the spherical 
of parallel, may be 
idians and parallels 
urface, and, more- 
the new surface 
cal surface, writing 
id kind; or rather, 
> 1. These values 
3 in each case a 
the equator), the 
In the first case, 
we may in place 
of the quadrilateral ABCD consider the birectangular triangle ABE; the new form of 
this is A'B'E', where the radius OA', = k. OA, = k, is less than the original radius 
Fig. l. 
unity, but OE\ — EJc, is greater than the same radius unity. The surface has at E' 
a conical point, the semi-aperture of the cone being =tan _1 ^- / , if as usual k' — V(1 — k*); 
k 
to verify this, writing for convenience q = 0, we have for the meridian section ¿c = cos p, 
z = E (k, p), and thence ~ = — S ^ n ^ which for p = 90° gives ^ = — k'. 
dz sin p ° ax 
Observe also that for p = 0, = oo, viz. the surface of revolution cuts the plane of 
the equator at right angles. 
There is no limit to the arc AB, it may be = 360°, viz. we must in this case 
cut the hemisphere along a meridian to allow of the deformation; or it may exceed 
360°, the hemisphere spherical surface being in this case conceived of as wrapping 
indefinitely over itself, and we may instead of the half lune E'A'B', consider the lune 
included between two meridians extending from pole to pole, and therefore the whole 
spherical surface, conceived of as wrapping indefinitely over itself; the result is, that 
this may be deformed into a surface of revolution, which, in its general form, resembles 
that obtained by the revolution of an arc less than a semi-circle round its chord; 
the half-chord being greater, and the versed-sine less than the radius of the original 
sphere. 
If k > 1, there is obviously a limit to the latitude AD, =BC, of the spherical 
quadrilateral; viz. this is equal to sin“' 1 ~. Supposing that in the quadrilateral ABCD 
(fig. 1) the latitude has this limiting value, then (see fig. 2) the new form is 
A"B"C"D", where along the bounding arc C"D" the tangent plane is horizontal; viz. 
as before ^ —& sin p) _ q f or p — sin -1 ^. It is to be observed, that the 
dx sin p fc 
radii for the parallels A"B" and C"D" are k and k cos p respectively; the difference of