966 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



the terms of the two lowest orders to zero, to equations of the type (3), and (l) then 

 follows from (4). 



The cases of the functions G and H may be treated similarly. 



Thus the conditions at a fixed end are, to a first approximation, 



az 

 G + G_i = 0, |^(G + G_i) = 0, 



(6) 



We may add the exact condition 



n + n_i = o (7) 



The above argument may easily be extended so as to show that when the 

 displacements at the two ends are of order 0, and no force is applied except at the ends, 

 F, G, and 11 are in general of order- 1, and H of order 0. 



Order of magnitude of tlie strains at the ends due to the neglected transitory 

 modes. 



The approximate theory which has now been sketched gives an amply sufficient 

 account of the nature and magnitude of the strain at any point of the cylinder not 

 very close to an end. It is not difficult to make out certain definite results with 

 regard to the order of magnitude of the strains at the ends themselves. 



Consider, e.g., the case of a rod built in at both ends, and subjected to any given 

 load producing za^-flexure. 



As was stated at (l), and has just been explained, the particular solution defined 

 byF_2, F_i, etc., can be approximately balanced at the ends. The residual unbalanced 

 displacements, and therefore also the displacements of the balancing transitory modes, 

 are of an order higher by two than F or F_2. Hence the strains at the ends due to the 

 transitory modes are of an order higher by one only than F or F_2, and are therefore 

 of the same order as the chief strains due to F and F_2, viz. e^^, eyy, e,^. That the strains 

 at the ends due to the neglected part of the exact solution may be as important as 

 those given by the approximate theory is a conclusion which should not be overlooked 

 in any application of the theory to questions of strength. 



48. Finite Cylinder. Independent investigation of the form of the asymptotic 

 .solution. 



In Art. 38 we collected the leading terms of the solution for body force and surface 

 traction in an infinite cylinder. These are of course the leading terms of an exact 

 particular solution for the same force in a finite cylinder. In Art. 43 it was shown 

 that the supplementary terms required for the exact solution of any problem for a 

 finite cylinder are made up partly of terms of the permanent type, as in Art. 26, and 

 partly of transitory terms the influence of which is confined to the neighbourhood of 



