896 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



So far these solutions are only improvements and extensions of Pochhammer's, but 

 at this point a step is taken which completely alters their character, and makes them 

 immensely better fitted for application to the Approximate Theory of a finite cylinder. 

 This step is to transform the definite integrals which occur in the general term of the 

 solution into infinite series, by means of Cauchy's Theorem in the Theory of Functions 

 of a Complex Variable. The solution is thus expressed in terms of a doubly infinite 

 series of particular solutions of the general equations of equilibrium under no forces. 

 The form of the general term is discontinuous at the cross-section which passes through 

 the source, that is to say, the solution is expressed by series of one form on one side of 

 the source, and another form on the other side. 



The particular solutions of which the general solution for a single force is composed 

 fall into two distinct classes, on whose characteristic properties immediately depends 

 the peculiarly simple form which the solution of the most general problem approximately 

 assumes. 



The first class we call the permanent free modes of equilibrium of the infinite 

 cylinder. These involve only rational integral functions of the rectangular coordinates 

 of a point of degrees up to the third. They consist of the six modes of equilibrium 

 discovered by St-Venant, and of the six rigid body displacements. 



The second class we call the transitory free anodes. Each of these is defined in 

 terms of a harmonic function of the type 



e -J m — COS m\<ii - oio), 



where /3 is a root of a certain transcendental equation which does not involve a, the 

 radius of the cylinder. This equation has an infinite number of real, and an infinite 

 number of complex roots. The roots occur in pairs diff"ering in sign only, and only 

 such of the transitory modes occur in any solution as make ^z negative, so that this 

 part of the solution vanishes at an infinite distance. Obviously, when 2 is a moderately 

 large multiple of a, the value of the above function is very small, so that the influence 

 of the transitory mode is very slight, and the displacement and strain are approximately 

 given by the permanent modes only. 



The coefticients with which the various permanent modes occur are rational integral 

 functions of the coordinates of the source of the third and lower degrees. There are 

 simple and interesting reciprocal relations between these coefiicients as functions of 

 {x\ y', z') and the displacements of the permanent modes themselves as functions 

 of {x, y, z). 



The strains due to the permanent modes depend only on the statical value of the 

 applied force, a striking example of St-Venant's principle of the elastic equivalence 

 of equipollent loads. But the rigid body displacements which occur in the source 

 solutions cannot be neglected, as they are not the same on the two sides of the source, 

 and, when solutions come to be deduced for extended distributions of force by 

 means of integration, the contribution of these rigid body displacements to the 



