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undergo in consequence of the displacement of the surface. 
From these expressions Gauss’s celebrated theorem as to the 
product of the principal radii of curvature follows at once. 
The author has next proceeded to consider the effect of 
fixing any curve, or portion of a curve, upon the surface. In 
this investigation it is necessary to consider severally the three 
classes into which surfaces are divided with respect to their 
curvature, namely— 
1. Oval surfaces, or those in which the principal curvatures 
have the same sign. 
2. Developable surfaces, in which one of the principal 
curvatures vanishes. 
3. Concavo-convex surfaces, in which the principal curva- 
tures have opposite signs. 
The author has obtained the following remarkable results : 
1. Ova SuRFACEs. 
If a curve, or portion of a curve, traced upon an oval surface 
composed of an inextensible membrane, be rendered immovable, 
the entire surface becomes immovable also. 
2. DEVELOPABLE SURFACES. 
I. Tf an are of a curve (which is neither the aréte de re- 
broussement, nor one of the rectilinear generatrices) traced upon 
a developable surface, be rendered immovable, all that part of the 
surface which lies between the rectilinear generatrices, drawn 
through the extreme points of the fixed curve, and on the same side 
of the aréte de rebroussement with the fixed curve, will become 
immovable also. Beyond these limits the surface will have the 
power of motion. 
II. The aréte de rebroussement, or a rectilinear generatriz, 
may in general be fixed without rendering any finite part of the 
surface immovable. 
Ill. The rectilinear generatrices of a developable inextensible 
surface are rigid. 
3. Concavo-convEX SURFACES. 
Tn surfaces of this class there exist (as is well known) at 
