AND DEFOK.MATIoN iK A THIN ELASTIC SHELL. Iti 



these suppositions are more general than those of the former paper, and enable the 

 author to take account better of the effects of aeolotropy of the material of the plate. 



DE ST. VENAXT* obtains the equations for flexure on the assumptions (I) that 

 R = 0, (2) that the middle-surface of the plate is bent without stretching, so that 

 the extension of any line-element through a point distant z from the middle surface 

 and parallel thereto is z/p, where p is the radius of curvature of the normal section of 

 the middle-surface through a line parallel to the element. From these suppositions, 

 of which the first is justified in the manner of BOUSSINESQ'S memoirs, the ordinary 

 equations are deduced and extended to the case of eeolotroptc plates. From the 

 inapplicability of the second of these suppositions to the case when the plate is 

 initially curvedt we may be justified in denying it the right to be a foundation for 

 the theory. 



The question between the methods of KIUOHHOFF'S second theory and BOUSSINESQ'S 

 memoirs may be taken to be that of the degree of approximation obtainable by the 

 former. It seems to be established that the terms which occur in CLEBSCH'S equations J 

 are correct to the order of approximation adopted ; but the question arises whether, if 

 it were desirable to obtain a higher degree of approximation in the equations, this 

 could be effected by means of KIRCHHOFF'S second theory ; and it appears that, so 

 long as the equations S, T = are retained with R = for the purpose of giving 

 three of the strains in terms of the rest, this question must be answered in the 

 negative. It must be observed that KIRCHHOFF only uses these equations for this 

 purpose, just as BOUSSINESQ does in his first memoir, while the equations and con- 

 ditions are found by applying the principle of virtual work. 



In a recent paper I have proposed a modification of KIRCHHOFF'S second theory, 

 with the view of showing how his kinematical equations, whose accuracy has been 

 disputed by BOUSSINESQ, can be made exact. The equations referred to are those 

 unnumbered on page 452 of the ' Vorlesungen.' In these certain differential 

 coefficients are introduced, and afterwards neglected as small ; and BOUSSINESQ has 

 contended that they should be retained. In the paper referred to I have endeavoured 

 to show that these differential coefficients have no meaning so long as we are treating 

 the equilibrium of an elementary portion of the plate, all whose dimensions are of 

 the same order as the thickness, so that the equations can be made exact by simply 

 omitting these differential coefficients. As will hereafter appear, KIRCHHOFF'S process 

 applies directly to the theory of a thin elastic shell, and the modification proposed in 

 the theory of plates has place equally in that of shells. This will be fully explained 

 in the sequel (Art. 2). 



* Translation of CLKBSCH. Note to 73. 

 t This will be proved in the sequel. 

 J ' Elasticitat,' pp. 306, 307, equations (105) and (106). 



" Note on KIRCHHOFF'B Theory of the Deformation of Elastic Plates," ' Cambridge Phil. Soc. Proc.,' 

 vol. 6, 1887. 



