Flexure of a Flat Elastic Spring. 217 



a watch-spring) into a circle of radius comparable with a 

 third proportional to its thickness and its breadth." 



The difficulty thus arising in the case of a straight band, 

 when the flexure exceeds a certain amount, makes its 

 appearance ab initio, even for infinitely small changes of 

 curvature, in the case of a band originally curved, and is in 

 this way closely connected with the circumstance, first 

 pointed out by Mr. Love,* that it is in general impossible 

 to satisfy the boundary conditions for a curved plate or 

 shell by a deformation such that the middle surface is 

 absolutely unextended. In considering this latter question 

 I had occasion, recently, - ]* to work out the uniform flexure 

 of a cylindrical plate, but I did not notice at the time that 

 the same analysis, with the proper change of meaning of a 

 coefficient, gives the solution of the problem proposed by 

 Thomson and Tait. As the matter is of independent 

 interest, and as so much importance has been attached to it 

 by these writers, I take the liberty of reproducing the 

 investigation (as suitably modified) in a separate form. 

 The main results are such as might be anticipated from the 

 above quotation. The extensions and contractions of the 

 middle surface of the band, which are called into play by 

 the tendency to the contrary curvature in the direction of 

 the breadth, keep this curvature in check, so that the 

 strained form never deviates appreciably from that of a 

 cylinder. When the radius p is large compared with the third 

 proportional aforesaid, the couple required to maintain the 

 flexure has the value given by the ordinary theory ; whilst 

 in the opposite extreme it tends to the value appropriate to 

 a plate. For intermediate cases we must have recourse to 

 the general formula (18) given below. 



Considering, then, a straight flat spring whose breadth 



* Phil. Trans., 1888 (A), pp. 521, 524. 



t Proc. Loud. Math. Soc, Jan. 1890. See also §7 of a previous paper 

 "On the Flexure of an Elastic Plate," Dec, 1889. 



