484 X. STATICS OF DEFORMABLE BODIES. 



Now by Green's theorem, 137, 55), 



ff 



=v ds ~ 7A Vdx dy ' 



and both integrals on the right vanish. Accordingly, as in the 

 demonstration of Dirichlet's Principle, 



|^ = 1^ = 0, V = & - & 2 = const 

 ex cy 



But we have the condition & = 0, so that the difference V is zero, 

 and the solution is unique. 



If the contour is a circle, the solution is immediate, by the aid 

 of circular harmonics. For the function & being developed in a 

 series of such, 



103) &=(A n r n co$n(p -f B 



, o 

 we have 



104) ? = =nR n -\A n cosncp 



o 

 so that if f(Xj y) is given developed as a trigonometric series, 



105) f(x, y) =(C n cos ny + D n sin <p) 



we must have, 



. 



which determines &. 



It may also be proved by the principles of the conformal 

 representation of plane areas, that the problem can be solved for any 

 contour whose area can be conformally represented upon a circle. 



Let us now put 



107) a^pr+bM + b 2 v 2 , 



so that if in the whole cross -section, 



108) z/J^=z/F 2 = z/F 3 = 



and on the contour, 



dV 



^ = y cos (nx) + x cos (ny)j 



109 ) 5 - [I x * + (* - 1) y'] cos ^ + & + $ *y cos ^y^ 



= (2 + ,) xy cos (nx) + ly> + (l- * 2 cos (ny) , 



