EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 963 



(xi)=-l^;.a*|(n + n_i), 



(XII)=-l:r/.aXn + n_i). 



Interesting relations may be obtained from these by elimination of F, G, H, or IT 

 by ^-differentiation, e.g. 



(III) + |-(lV) = l7rEa4'^i 

 dz 4 dz 



-z 





a^x > 



All the functions (I) to (XII) are continuous functions of z, even when the applied 

 force is discontinuous at the section z. This follows either from the explicit values of F_2, 

 etc., given in Art. 38, or from the fact that the transitory modes contribute nothing 

 to the integrals. We thus obtain another solution of the problem of Art. 44, not 

 differing essentially from the solution given there, but somewhat simpler from the 

 physical point of view. 



46. Another method for calculating the twelve integrals. Modified form of the 

 twelve relations. 



The values found for the integrals (I) to (XII) in last Article can be obtained 

 immediately with the help of Betti's Theorem. 



In each of the permanent modes of Arts. 21 and 22 write z' — z instead of z, and 

 take X, y, z', instead of the usual x, y, z as current coordinates. For the application of 

 Betti's Theorem take for first system one of the permanent modes (as just modified), 

 and for second system the deformation [u^, Uy, u^, zx, zy, zz) due to body force X, Y, Z, 

 with cylindrical tractions X^, Y^, Z^, and end tractions. 



Apply the Theorem to the slice between z' = z^ and z' = z. 



Thus from the mode of Art. 21 I. we get, for the integral (IV), at the section z. 



i i_(/ - zf + (z' - z){ax^ - cry'- - m^) | X + 2ct{z' - z)xyY 



III 





+ -(^1 



.,)2,r+^:,(.r2+y2y 



0- + - 



a% > zz 



_ - /. I ai:,? - y2) + |_.,2 + ^_y2 _ ^^ + ^y I ,,^. _ ^(^^ + \yxyu,^ + 2E(2, - z).ru, \ __ 



JS,„ (1) 



z 



2 being defined as in Art. 38 (l), (2), and the notation 2 [] being used to indicate 

 that the values in [ ] are taken at the section z'. 



The second integral on the right of (1) is a rational integral function of z of the 

 third degree ; and the first term of (l) obviously differs from + ;^7rEa*(F_2 + F_i + F,, + Fi) 

 by another function of the third degree. 



Hence the result agrees with the value of (IV) given in Art, 45. 



The remaining integrals of (I) to (XII) may be dealt with in the same way. 



