EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 929 



of the infinite cylinder. They might be taken to be their vanishing at one end, but 

 the course actually adopted has certain advantages connected with symmetry. (See 

 Art. 20 before equation (1).) 



It may be noticed that the 2-derivatives of the displacements of the fourth and 

 higher orders vanish at infinity ; we may say, in a certain sense, that infinity is an 

 accidental, not an essential, singularity of the solution. 



The force {X, Y, Z) at {x', y' , z') is statically equivalent to a force (X, Y, Z) at the 

 origin and a couple {y'Z — z'Y, z'X — x'Z, x'Y — y'X.). 



If we consider the equilibrium of a portion of the cylinder bounded by two sections 

 on opposite sides of the source, we see that the stresses at the two ends contribute 

 equally to the balancing of the force at {x', j/', z'), the permanent displacements at 

 the two sides of the source diff"ering in sign only, and the transitory deformation 

 contributing nothing to integral stress. 



Hence the solution, S say, for the force (X, Y, Z) at {x, y', z') will contain, for z>z', 



X . S, + Y . 83 + Z . S3 + (y'Z - zY)S^ -f (z'X - x'Z)^^ + {x'Y - y'X)8^ ; \ 

 and in addition > . . . (2) 



ttySy + OgSg + ngSg + ttjoSjo + ajiSii + °-iS\2 > ) 



where a^, . . ., a^^ are coefiicients to be determined. 

 For z<iz' , S will contain 



-X.Sj- .... -(a;'Y-2/'X)S6, ) 



and V (2)' 



— a^o^ — .... — ajr.Sj^Q • / 



In either region, the remaining part of the solution is made up of transitory 

 modes only. 



Let the displacements and z-stresses belonging to the solution S, be denoted by 



u}'\ u,j'\ M,"'' ; zS'*-', zy'-'\ zz'-'-\ 



Also let 



W(r,s)s/" f(M/S''''' + M/'5p<^* + w/-'£''''-w/'2BW-w/'zy^ . . • (3) 



taken over a section of the cylinder ; by Betti's Theorem the result is the same for 

 every section and may be conveniently calculated for 2 = 0. 

 For any two systems of displacements, we call 



I I (displacements of first • stresses of second - displacements of second • stresses of first)(f S 



the work difference of the first and second systems for that section ; evidently it 

 changes sign when we interchange the two systems. 



Now for first system take the solution S for the force (X, Y, Z) at {x' , y' , z'), and 

 for second system the permanent mode S^, r=l, 2, . . . 6, and let the part of the 

 body considered lie between the planes z^z-^ and z — z^, where z^<z!<z.-^. 



The work difference of S, and any of the transitory modes is zero, since it is 

 independent of the section considered and vanishes at infinity. 



