928 DR JOHN DOUOALL ON AN ANALYTICAL THEORY OF THE 



The integral stress is a 2a;-couple of amount 



I I (zzx - xzz)dS = — TrEa^. 



VI. 



S5=(this solution)( — ^^r-A 



zx= - jxy, zy = fix, zz = 0. ) 



pp = 0, pco = 0, pi = 0, wu) = 0, Sz — fip. ) 



Tlie integral stress is an ic^z-couple of amount 



// 



(pczy — yzx)dS = — Tr/^a*. 



Sgs(this solutton)f - ^ 



22. Verification of the coefficients of the permanent modes in the solutions for a 

 single force, by means of Betti's Theorem. 



Let a force with components X, Y, Z be applied at a point {x', y', z') in the body 

 or on the surface of the infinite cylinder. 



We have seen in the preceding pages that a solution exists consisting of a 

 permanent and a transitory part, such that if in the region z^z' the form of the 

 permanent part is 



then its form in the region z<z' is 



In either region the permanent part of the solution is compounded of certain 

 multiples of the solutions Sj, Sj, S3, S4, S5, Sg of Art. 21, and of the six rigid body 

 displacements 



^7 - ■ '^^y = 0> Sg = <[ ?<, = 1, %=\u,j = Q, 



\ u^ = 0. \ u^ = 0. \ u,= l 



I «^ = 0, / M^ = z, . w^= -y, 



(1) 



' u,= y. ' M^= -a;. . ' m, = 0. 



Seeing that we might add to a solution of a given problem any multiples of S,, S2, 

 . . . S12 without affecting the body or surface conditions, it is not necessary to assume 

 that the permanent displacements in the two regions 0>2' and z<Cz' are the same except 

 for sign ; but, unless some such condition is laid down, all that can be determined is the 

 difference of the coefficients with which any particular mode occurs on the two sides of 

 the source. The solution is not determinate in the absence of conditions at infinity. 

 These cannot be taken to be the vanishing of the displacement and stress at hotJi ends 



1 



