EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 931 



+ 



u. 



2(Txy, 



2(Txy, 



-2zx, 



{y'Z-zY 



rEa* 



(z'X - x'7.)( - -L \ 



■ zx, 

 = 0, 



= 1/ 



= 0, 



:0 



{xY-,j'X) 



TTfia' 



■ + z'{o-(a^'2-y/2)-T,/2] 



X 



~u^ = 0~ 





Uy=1, 





_M, = 0,_ 





+ 2(TZx'y'Y 



+ .j - 2'2a;' + lx'{x'^ + 2/'2) - ('o- + A - tVvIz 



2o-2'a;'y'X 



+ { - ^'Y + ^2/'(-«'' + y'-') - ('^ + 1 - ^)" Y'}z 



TrEaV 



rEa* 



^0, 

 0, 



1, 



= 0, 



= -z, 



■y, 



--Z, 

 -0, 

 ■■ -X, 



= -y, 



--X, 



= 0, 



— (tx X 



- o-//'Y 

 + z'Z 



1 



27rEa2 



2o-a;yX 



+ {2'2 + (r(2/'2-x'2)}Y 



_ - 2.'2/'Z 



"{z'2 + o-(a;'2-t/'2)}X' 

 + 2<Tx'y'Y 



- 22VZ 



TrEa^y 



1 



rE«4 



-^yx 



+ z'aj'Y 



1 



iV.5. — The coefficients of X, Y, Z in the cofactors of each of the twelve modes are 

 respectively proportional to the u^,Uy,U2 of the conjugate mode in accented co- 

 ordinates. {Cf. end of Art. 22.) 



24. Verification from the preceding table of certain reciprocal relations deducihle 

 from Betti's Theorem. 



These results admit of an interesting verification by means of certain reciprocal 

 relations analogous to the well-known reciprocal property of the Green's function in 

 the Theory of the Potential, viz. that the Green's function at P for a pole at Q is 

 equal to the Green's function at Q for a pole at P. 



Consider two deformations of the infinite cylinder with free surface and subject to 

 the symmetrical conditions at the two infinitely distant ends laid down in Art. 22 



