494 MR. A. E. H. LOVK ON THE SMALL FRKK VIBRATIONS 



plate ; and he says that R = to a higher degree of approximation than S and T. 

 Taking 2h for the thickness of the plate, and the plane of xy for the middle -surface 

 in the natural state, we have, on integration, with reference to 2, 



hfUS 



dc dy 



+ ?+.)* 



a. ci/j 



Assuming that the bodily forces are not such that if- applied to a body of finite 

 size they would produce deformations indefinitely great compared with those produced 

 in the plate, and that P, Q, U do not vary rapidly from one element to another, we 

 see that S, T, R are small compared with P, Q, U. BOUSSINESQ proceeds to express 

 three of the strains in terms of the rest by means of the relations S, T, R = 0, as 

 was done in KIRCHHOFF'S second theory ; then, by means of these approximate values, 

 he finds S, T to a higher order, and on substituting in the general equations of equili- 

 brium obtains the well-known equation for the deflection of the plate. The method 

 of securing the union of two of POISSON'S boundary-conditions in one is the same as 

 that previously given by THOMSON and TAIT. 



BOUSSENESQ returned to the subject in 1879, in a second memoir published in 

 ' LrouviLLE's Journal.' Apparently dissatisfied with the assumptions S, T = 0, he 

 proposed to consider the subject in the following manner. Let the plate be divided 

 into similar elementary rectangular prisms, whereof the linear dimensions are all 

 comparable, and suppose these prisms bounded by the plane surfaces of the plate, and 

 by pairs of parallel planes at right angles to these surfaces. Two neighbouring 

 prisms must always be in nearly the same condition as regards strain, except in the 

 case of prisms situated near the edge. Hence, generally, the component stresses will 

 be approximately the same at all points on the same surface parallel to the middle- 

 surface, and not infinitely near the edge of the plate. Hence, in this kind of 

 equilibrium, the stresses will be approximately independent of the position on the 

 middle-surface of the centre of the element. This is precisely KIRCHHOFF'S result* 

 deduced from the kinematics of the system, and it appears certainly true when the 

 plate is very thin. BOUSSINESQ wishes his theory to apply to plates of small finite 

 thickness, and he proposes to replace the equations just found by the following 



T. 8T 8f 8S 8S ' ,,,. ~ TT , 



' Vorlesungen,' p. 453, remark on equations (8). 



