106 Dr. J. Prescott on the Equations of Equilibrium 



true near the edge of the plate. The best physical explan- 

 ation o£ this difficulty has been given by Professor Lamb in 

 his paper "On the Flexure of an Elastic Plate" ^Proceedings 

 of the London Math. Soc. vol. xxi.). 



Our new differential equations may require new boundary 

 conditions. I think, however, that it is unlikely that we 

 can satisfy more than the four boundary conditions of 

 Kirchhoff, though this number depends upon the number 

 of arbitrary functions, involving only real quantities, that 

 are contained in the general values of w and (/> satisfying 

 equations (12) and (14). How many such functions there 

 are I have to admit that I am unable to discover. Although 

 some rules are given in the theory of partial differential 

 equations concerning the number of arbitrary functions in 

 the solution of a partial differential equation, these rules are 

 not, I think, of any use for our purpose, because they make 

 no distinction between functions with real arguments and 

 functions with imaginary arguments. The theory tells us, 

 for example, that the solution of the equation 



contains two arbitrary functions. True, it does. But when 

 we are confined to real quantities, these only amount to one., 

 because the solution 



Y =/(# + %y) + FO— iy) 



degenerates into 



Y =/(# + iy) +f(% — iy) • 



I believe, however, that the number of boundary conditions 

 is four in all cases, because it seems to be definitely four in 

 some cases. Suppose, for example, that an unstrained plate 

 has its edge completely fixed and clamped in its unstrained 

 position. Then let any transverse forces be applied to the 

 plate while the edge is still held. The problem in this case 

 appears to be completely determinate, and the conditions at 



the edge are u = 0, v=0, iv = 0, -^- =0, dv being an element 



of the outward normal to the edge. This suggests that the 

 functions iv and <£ are completely determined by the differ- 

 ential equations and four boundary conditions in one case, 

 and therefore also in all other cases as well. 



If we could solve equations (12) and (14) completely, we 

 should have no trouble in deciding how many boundary 

 conditions could be satisfied. But these equations are 



