500 Mr. A. B. Basset on the Finite 



Now, according to the fundamental hypothesis of my 

 former papers it follows that, provided there is no external 

 pressure, fi must be a quadratic function of h and 77, and con- 

 sequently the retention of E, will lead on integration to 

 terms in G 2 of a higher order than A 3 , which are to be 

 neglected, since the solution we require is an approximate 

 one which does not contain higher powers of h than the cube. 

 Accordingly if we substitute the values of o^, <j 2 from (2) and 

 (3) in (1), and the resulting value of P in (4) and integrate, 

 we shall obtain 



_ ^=i nh3 {^<h-}>^-})}- • (5) 



Similarly, 



Gr 1 =— I Qvdrj, 

 J-h 



which gives 



(6) 



Equations which are equivalent to (5) and (6) have been 

 given by more than one writer on elasticity ; but attention 

 has not always been called to the fact that they depend upon 

 the express conditions that the surfaces of the shell are free 

 normal pressures, and also that the extension of the middle 

 surface may be neglected. 



When a plane plate is bent into a developable surface 

 p 1 ~p^ = oo ; also one of the quantities p/, or pj (say p 2 ') is 

 infinite ; whence (5) and (6) become 



G 2 =f«A 3 (l +B)/ / > x '- l 



G 1= -fnA 3 E/ Pl ' )' • • .• • • ^ 



where G 2 is the couple about a generating line of the develop- 

 able. 



Since the extension of the middle surface is neglected, 

 equations (5) and (6) would not apply to the case of a plane 

 plate deformed into a surface such as a portion of a sphere. 



7. As an example of the preceding method, we shall con- 

 sider the case of a plane plate of thickness 2h, which is 

 bounded by two radii CA, CD, and two arcs OB, AD of 

 concentric circles ; and we shall inquire whether it is possible 

 to bend this plate into a portion of a right circular cone in 

 which OA, BD are generators, and OB, AD are circular 

 sections. 



We shall assume for trial that the bending may be effected 

 by means of tensions, normal shearing stresses, and flexural 



