960 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



found. The permanent modes are also easily balanced in these cases. For example, 

 for problems (6) and (c), the modes of Art. 19 corresponding to flexure in the plane zx, 

 viz. (] ) (ii) and (iii) with 5 (ii) and (iii), are balanced by a deformation of the type 



Art. 26 (i) ; in {h) F and g- i" (<^) -7- ^^'^^ "7-3" ^^'^ given at the two ends, and F is 



completely determined. A somewhat simpler method for these permanent modes will be 

 explained in Art. 47 (h), (c). 



The problems of given end tractions and of given end displacements do not admit 

 of such simple solutions. But we have just seen that when the end tractions are given 

 the six chief permanent modes can be found exactly. And in the important case when 

 the end displacements are given to be null, under a given internal force, the permanent 

 modes can be found approximately, e.g. in Art. 26 (i) we can take the values of F and 



r/F 



-J- at the ends so as to neutralise the principal values of the zx-flexural displacements.* 



44. Given particular solutions for the body and cylindrical surface forces in 

 separate regions hounded by cross-sectiofis, to determine for each region the subsidiary 

 free modes necessary for continuity of displacement and stress. 



It sometimes happens in cases when force is applied only to a limited portion of the 

 body that a particular solution for this portion is easily found. For example, the forces 

 may be those of Art. 3(1) between z — z^ and Z — Z2, and may vanish when 2 lies beyond 

 those limits. In this case the solution of Art. 3 is a particular solution for z-^<:iz<Cz2, 

 while 7iuU displacement is a particular solution beyond this region. We know from 

 the preceding Article that the complete solution can only difl"er from the aggregate of 

 the particular solutions by series of permanent and transitory free modes, these having 

 different coefficients in the separate regions, and the transitory modes having their 

 maximum influence at the section of discontinuity. 



One method of completing the solution would be to deduce the applied forces from 

 the fundamental equations of equilibrium and then solve the problem afresh as in the 

 preceding pages. The solution thus found would, for z^<Cz<Z2, diff"er from the given 

 particular solution by definite free modes. Subtracting these from our solution in 

 every part of the body, we find the subsidiary free modes in the exterior regions. 



But there is a simpler method depending on the properties of the mutual ^vork- 

 difference of two free modes (Art. 22), 



In the first place, the work-difference of two deformations at a given section is 

 continuous across the section, since it involves only the displacements and stresses. 



Next, to any free mode there corresponds a second, or conjugate mode, such that 

 the work-difterence of the two does not vanish, while the work-difference of either and 

 any other than its conjugate does vanish. (Art. 22 between (5) and (6), and Art. 27.) 



Considering now two regions separated by a section z = Zq, let 2j be a particular 

 solution on the side z<Zq, and ^2 01^ the side z'>Zq. 



* df. Art. 47. 



