

Flexure of a Flat Elastic Spring. 183 



and greater shortenings and stretchings in the filaments 

 towards the corners than those expressed in our formulae, 

 and so vitiate the solution. Unhappily, mathematicians have 

 not hitherto succeeded in solving, possibly not even tried to 

 solve, the beautiful problem thus presented by the flexure of 

 a broad very thin band (such as a watch-spring) into a circle 

 of radius comparable with a third proportional to its thick- 

 ness 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 un- 

 extended. In considering this latter question I had occasion, 

 recently f, 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 ex- 

 tensions 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 com- 

 pared with the third proportional aforesaid, the couple re- 

 quired 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 

 2b is large compared with the thickness 2/i, let us in the 

 first instance suppose that by a proper application of force 

 to the two ends (to be afterwards determined) it is bent so 

 that the strained form is one of revolution. As regards 

 the amount of the bending, we shall suppose only that p 

 the radius of curvature of the medial line or axis, is large 



* Phil. Trans. 1881 (A), pp. 521, 524. 



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

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



P2 



