ON THE FLEXURE OF A SPHERICAL SURFACE. 
[From the Messenger of Mathematics, vol. vi. (1877), pp. 88—90.] 
It is known that an inextensible spherical surface, or to fix the ideas the spherical 
quadrilateral included between two arcs of meridian and two arcs of parallel, may be 
bent in suchwise as to be part of, a surface of revolution, the meridians and parallels 
of the spherical surface being meridians and parallels of the new surface, and, more 
over, the radius of each parallel of the spherical surface being in the new surface 
altered in the constant ratio k to 1. We have, in fact, on the spherical surface, writing 
p for the latitude and q for the longitude, and the radius being unity, 
x = cosp cos q, 
y = cos p sin q, 
z =sin p, 
values which give 
dec 2 + dy 2 + dz 2 = dp 2 + cos 2 p dq 2 . 
This last equation is satisfied by the values 
q 
x = cosp cos ~ , 
• 9 
y = cos p sin ^ , 
z — E {k, p), 
where E (k, p), = I V(1 — k 2 sin 2 p) dp, is the elliptic function of the second kind; or rather 
J o 
this is so when k < 1, but the same notation may be used when k > 1. These values 
give the deformation in question. 
The two cases to be considered are k < 1, and k> 1; we take in each case a 
spherical quadrilateral ABGD (fig. 1), bounded by AB (an arc of the equator), the 
arc of parallel CD, and the two arcs of meridian AD and BC. In the first case, 
there is no limit to the latitude AD, = BC, or taking these = 90°, we may in place