498 Mr. A. B. Basset on the Finite 



involves a case of finite bending; but as its cross section is 

 approximately square, the theory of the bending of wires* 

 would be more applicable. Similar examples, such as spring 

 balances and other mechanical appliances where springs are 

 employed, will readily suggest themselves; and the question 

 whether the theory of wires or the theory of thin shells is 

 most appropriate depends upon the nature of the spring. If 

 the cross section does not differ much from a circle or a 

 square, the former theory would appear to be the most applic- 

 able ; if, on the other hand, the breadth of the spring is 

 considerable compared with its thickness, it would be better 

 to regard it as a thin shell. 



When the natural form of a spring is a plane curve, and 

 the spring is bent into another plane curve, the problem may 

 be completely solved by the methods explained in chapter viii. 

 of my 'Elementary Treatise on Hydrodynamics and Sound ' 

 The mathematical treatment is the same whether the spring- 

 be regarded as a wire or as a thin strip of metal like a clock- 

 spring; the only difference being that the flexural rigidity is 

 different in the two cases. If, however, a piece of clock- 

 spring is twisted as well as bent, or a thin plate or shell is 

 deformed in a finite manner, the solution of the problem pre- 

 sents difficulties of a rather formidable character. 



4. Whenever the deformation is finite, the displacements of 

 a point on the middle surface are not small quantities whose 

 squares and higher powers may be neglected, and therefore it 

 is useless to attempt to express the stresses in terms of these 

 quantities ; but since any deformation involves a change in 

 the values of certain geometrical quantities, such as the cur- 

 vature and torsion of certain lines drawn on the middle surface, 

 the most appropriate course to pursue would be to endeavour 

 to express the stresses in terms of such geometrical quantities. 



There is one class of problems which can often be solved 

 without much difficulty, which occurs when a plane surface 

 is bent without extension into a developable surface ; or when 

 a developable surface is bent into a plane, or into some other 

 developable surface such that the lines of curvature on the 

 old surface are lines of curvature on the deformed surface. 

 This method can generally be applied when a plane plate is 

 bent into a conical or cylindrical surface ; but it could not be 

 applied in the case. of a right circular cone which is bent into 

 a cone whose lines of curvature are not identical with those 

 of the former cone. 



The success of this method, in cases where it can be applied, 

 depends upon the circumstance that the flexural couples G 1? 

 * See Proc. Lond. Math. Soc. vol. xxiii. p. 105. 



