Certain Problems of Two-Dimensional Physics. 779 



Using the notation of the section last quoted, consider as 

 an example the problem o£ the bending of a plate by its own 

 weight, the edge being clamped in a horizontal plane. We 

 have, in this case, 



V 4 w = Z'/D = W/AD = 64a, say, 



where W is the weight and A is the area of the plate, with 

 the conditions 



w = "dw/df=0 



at the edge f = 0. The general solution is 

 w=fi{( J; 2 +^) 2 -KX 1 2 +Y 1 2 ) 2 -i(X/ + Y 2 2 )2 



+ 2,i e (X 2 + Y 2 )(XY-XY)*;} 

 -\- i [£[XF'( v ) + YG'( v )]d v + .v[F(0)-F(T)] + , / [G(d)-G(r)]} 



where F and Gr are functions to be determined from the 

 conditions of finiteness, &c. 



A sufficient illustration will be furnished by the considera- 

 tion of the case of the elliptic boundary, for which we may 

 evidently take 



F(0)=X2(Asin0 + Bsin30), G(0) = -fl(Ccos0+Ecos30), 



where A, B, C, & E are constants which may be determined 

 by equating to zero the coefficient of £ — since f becomes 

 infinite at the centre when the ellipse is a circle — and the 

 value of dw/d£ when f = —A,. The equations thus obtained 

 lead to 



A= %b (a a + b 2 ), C = 2a(a 2 + b 2 ), 



2b(3a 2 4-b 2 )(a*-V) , 2 a(a 2 + 3& 2 )(a 4 — 5 4 ) 



3a 4 + 2a 2 /> 2 + 3/; 4 3a 4 + 2a 2 b 2 + 36 4 * 



On substituting these values in the expression for w, we 

 obtain after reduction the — otherwise obvious — result, 



w 



= (W/87rD)a 3 /> 3 (l- -■ - y ~X (3a 4 -f 2a 2 6 2 +36 4 ). 



The solution of the problem here given is, of course, to be 

 regarded merely as an illustrative example of the general 

 process. 



