1888.] Sending and Vibration of thin Elastic Shells. 121 



plane is a purely cylindrical one which leaves the middle generating 

 line straight. There are two ways in which we may conceive the 

 strip altered so as to render it susceptible of the desired kind of bend- 

 ing. The first is to take out the original cylindrical curvature, which 

 reduces it to a plane strip. The second is to replace it by one in which 

 the middle line is curved from the beginning, like the equator of a 

 sphere or ellipsoid of revolution. In this case the total curvature 

 being finite, the Gaussian condition can be satisfied by a change of 

 meridianal curvature compensating the supposed change of equatorial 

 curvature. It is easy fco calculate the actual stiffness from (8) and 

 (14), for here T = 0. We have 



Pi 



which expresses the work per unit of area corresponding to a given 

 bending Sft" 1 along the equator. If p 1 = co, the cylindrical strip is 

 infinitely stiff. If the curvature be spherical, p% = p^ and 



and if p 2 = oo, 



U = _^1. _^_ ffri-Y (49). 



3 m+n \ pi/ 



Whatever the equatorial curvature may be, the ratio of stiffnesses in 

 the two cases is equal to m : m + n, or about 2 : 3, the spherically 

 curved strip being the stiffer. 



The same principle applies to the explanation of Bourdon's gauge. 

 In this instrument there is a tube whose axis lies along an arc of a 

 circle and whose section is elliptical, the longer axis of the ellipse 

 being perpendicular to the general plane of the tube. If we now 

 consider the curvature at points which lie upon the axial section, we 

 learn from Gauss's theorem that a diminished curvature along the 

 axis will be accompanied by a nearer approach to a circular section, 

 and reciprocally. Since a circular form has the largest area for a 

 given perimeter, internal pressure tends to diminish the eccentricity 

 of the elliptic section and with it the general curvature of the tube. 

 Thus, if one end be fixed, a pointer connected with the free end may 

 be made to indicate the internal pressure.* 



* Dec. 19. It appears, however, that the bending of a curved tube of elliptical 

 section cannot be pure, since the parts of the walls which lie furthest from the 

 circular axis are necessarily stretched. The difficulty thus arising may be obviated 

 by replacing the two halves of the ellipse, which lie on either side of the major 

 axis, by two symmetrical curves which meet on the major axis at a finite angle. 



