98 Dr. J. Prescott on the Equations of Equilibrium 



The only previous reference to this limitation that I am 

 aware of occurs in Thomson and Tait's 'Natural Philosophy.' 

 Thev state (Art. 632) that " the deflexion is nowhere, within 

 finite distance of the point of reference, more than an 

 infinitely small fraction of the thickness/'' They had 

 already worked out (Art. 629) the mean circumferential 

 strain of a small circular portion of the middle surface of a 

 plate which is bent into a surface with principal curvatures 

 pi and p 2 when the radial lines are unstrained. From their 

 result they justify the limitation that they impose. 



The footnote added to the above quotation from Thomson 

 and Tait by Professor Karl Pearson in the ' History of the 

 Theory of Elasticity ' shows how little the point was under- 

 stood. The footnote runs thus : " The pressure of the finger 

 on the bottom of a round tin canister seems to produce a 

 deflexion which is far from being an infinitely small fraction 

 of the thickness, and which might, I think, be fairly discussed 

 by the ordinary theory." I regard this footnote, written by 

 the author of such a complete history of the modern develop- 

 ments of the subject, as very good evidence that no other 

 writers have laid any stress on this very important limitation 

 of the usual theory. 



We shall now produce a theory in which the only 

 restrictions are that the thickness of the plate and the 

 maximum displacement just mentioned are of smaller order 

 than the radii of curvature of the bent middle surface. This 

 will bring the plate theory to the same level as the Bernoulli- 

 Eulerian theory of beams. 



Let us assume that the xy plane touches the bent middle 

 surface of the plate at some point, and that a particle of the 

 middle surface which, in the unstrained state, would be at 

 x, y, 0, is displaced to x-\-u, y-\-v, iv. Let dx, dy denote 

 the components of an element of length ds in the unstrained 

 middle surface, and let ds 1 be the length of this element 

 after strain. Then 



(ds) 2 ={dx) 2 + (dy) 2 

 and {d Sl f = (dx + du) 2 ±(dy + dv) 2 + (dw) 2 . 



Now we may neglect (du) 2 and (dv) 2 since we shall retain 

 the more important quantities du and dv, but we have no 

 such reason for neglecting (dw) 2 . Then 



(ds ± ) 2 = (dx) 2 + (dy) 2 + 2dxdu + 2dydv + (dw) 2 



= (ds) 2 + 2dxdu + 2dydv + (div) 2 . 



