EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OP CIRCULAR SECTION. 913 



Z force of iTTfiv units at {x', y', z'). 

 For this Z force the displacements are * 



(x - x')(z - z\ , ,sj)r~^ 



^z 



r'' cz 



{z-z'fv-l 



,\ dr' 



- + ~:— == (•*'■ ~ ^') g^- + (y - .'/ 



w 



here 



9y 



+ vr' 



r''^ = (:x-j;f + {y-y'f+{z-zf. 



Comparing 2 (7) with (l), we recognise that the latter is compounded of 



(i) a strain of the (^ type in which -X = - ?'"\ 



(n) .M^ = u: — , u^^O, u,= -x~, 



(ill) 2<., = 0, u =ij u,= -2/-—. 



3z 9// ; 



Let DjV"^ stand for a harmonic function whose z-derivative is r"\ 

 We may put (2) (ii) in the form 



// = X — - D, V 



■<£-$p^-'- 





Kdxdy dxdi/J 



in which we see 



oxoz 



3x 



I = - x'r^J)7h'-\ ^= - jc'^D; V-i. 



■'A 

 3y' 



Similarly, we see that 2 (iii) is equivalent to 



,3. 



l=--,-^Dr.-, ,.,'^D; 



4-4)"^'-^' ^K4-4)^^^^-^- • 



(1) 



(2) 



(3) 



(4) 

 (5) 

 (6) 



Thus altogether (I) is equivalent to 



As definition we take here 



DrV-i = — / sin k(z - 2')Goi/cE -lcZ,c, (7) 



where E^ = (x - x')^ + (2/ - /f ; this is allowable, for the 2-derivative of the harmonic 



function on the right of (7) is 



2 /'" 



- cos k{z - z')QJ.KRdK, ....-■. (8) 



a well-known expression for r"\ 



We shall require also a z-integral of the function of (7), for which we take the 

 harmonic function defined by 



DrV-^= -~f {cos/c(z-3')G„z/<E-log2 + y + logKll}i-tZ'c. . . . (9) 



Love, Elasticity, Art. 130. 



