w 



jor Differential Equations of Mathematical Physics. 599 



powers of C equated to zero. It will be seen that there are 

 thus sufficient equations and boundary conditions for each of 

 the functions -^ ,^ l5 etc. to determine them uniquely. It is 

 not proposed here to prove the uniqueness of tliese expressions, 

 but rather to point to an analogous series of physical problems 

 with which each of these equations correspond. 



§ 4. Parallelism in the theory of elastic plates. Flexure o] 

 a flat plate. — If a flat plate of flexural rigidity EI per unit 

 width, Poisson's ratio a be loaded laterally with an intensity Z 

 per unit area, the plate being supported in any given manner 

 along the edges, then if -v/r be the deflexion at any point 

 measured relative to a given horizontal surface, the equation 

 determining -v/r is 



DV 4 f = Z, (15) 



here D = EI/(l-a ? ) and I = 2/.W, A= thickness of plate, 

 hile the shearing force N per unit length of any curve 



drawn upon the plate is in the direction perpendicular to 



the plate ; 



N=-D| n V 2 f, .... (16) 



when n represents the normal to the curve. 



A complete parallelism may now be established between 

 the problem of the flexure of a flat plate under lateral loading 

 and each of the problems involved in the determination of 

 the functions yfr , yfr u . . . , the boundaries of the flat plate 

 being the same as those of the fluid. 



For the evaluation of the function yjr in that case the 

 following equations and boundary conditions were required 

 to be satisfied : — 



V 4 fn = /(to, fi ■ • • f »-i) 5 n - 0, 

 r^ _ ^__ o a i on g every boundary. 



body ^body 



For the evaluation of ^ , 

 V 4 ^ = o, 



—^ = — 1, ?r^ = alone: the moving boundary. 

 and 



i -f (V 2/ ^o) ^- s " = round the body. 



bodv 



2R2 



