898 DK JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



permanent and transitory modes. The permanent modes are then found exactly in 

 tlie cases when the data at an end are one or other of the three sets, not necessarily the 

 same set at the two ends, viz. : — 



(i) B,zy,zz, 

 (ii) u^,u^,zz, 

 (iii) u,,£:,zi/. 



In cases (ii) and (iii) it is pointed out how the complete solution, including the 

 transitory modes, can easily be found. The case (ii), with u^, Uy, zz all zero, corresponds 

 closely to the important practical case of a simply supported end. 



When the data at an end are 



(iv) %. ,u^,u^, all zero, 



corresponding to a built-in end, the permanent modes are deduced to a first 

 approximation. 



In the aggregate result, it is completely confirmed that the ordinary approximate 

 theory does really give a first approximation to the exact solution. 



Finally, an independent investigation of the asymptotic solution is given, based 

 directly on the difi"erential equations which hold in the body and at the cylindrical 

 surface. This investigation brings out very clearly from one point of view the origin 

 of the peculiarly simple form of the leading terms. This is seen to depend essentially 

 on the fundamental principle that, unless the applied forces fulfil of themselves the 

 statical conditions of equilibrium, the only solution of the differential equations is a 

 rigid body displacement. An analysis of the asymptotic solution has been given by 

 PocHHAMMER.* The interest of his investigation suff'ers somewhat, however, from its 

 being based on the initial assumption, not completely justified, that zz is of lower 

 order than any other component of stress. 



The work on this paper was undertaken in connection with my appointment to a 

 Carnegie Research Fellowship. I am glad to have the opportunity of acknowledging 

 my great indebtedness to the Carnegie Trustees for their generous assistance, but for 

 which it is not at all likely that I would have been able to complete the paper. 



1. Equations of equilibrium. Strains and stresses in cylindrical coordinates. 

 The equations of equilibrium of a homogeneous isotropic elastic solid in rectangular 

 coordinates are 



Zxx dxy dxx ,^ 

 3-+^ + -5- + X = 0, 

 ox ay oz 



arp SJ/ 3^2 



dxz dvz dzz 



(1) 



where X, Y, Z are the components of the body force per unit volume ; and xiJt, yy, zz, 



* L. PoCHHAMMER, Uutersuchungen nber das Gkichgemcld dcs elastinchen Utahes, Kiel, 1879. 



