946 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



(for the series is uniformly convergent if e>0), 



= 2Li ) a mean of cos i-^ '-f.2{z ) ^ s' e « - e « V , . (3) 



This can be split up into two series corresponding to the two terms of the last 

 bracket. The limit for e = of the second series is obviously of " infinite order" in a. 

 As for the first series, its limit for e = we know to be finite (for we are dealing 

 with a displacement or stress), and by putting e = at], we see, as at the end of last Article, 

 that the z'-integration has increased the order by one. 



To sum up these results. 



Any component of the applied force is supposed to have the form ^{x/a, y/a, z), 

 and is therefore of order zero in a. 



We first of all calculate the displacements and stresses due to a single force. 

 Each displacement and each stress is composed of a permanent and a transitory part. 

 The rigid body displacements in the source solutions will contribute nothing to the 

 permanent terms in the stresses. 



(i) Body force. — The transitory displacements due to a unit force are of order a'^ 

 at 2 = z'. Integration over a cross-section introduces a factor a'\ and integration as 

 to z' a factor a. Thus the integrated transitory displacements are of order a^. 

 Similarly the integrated transitory stresses are of order a. 



(ii) Surface force. — The element of surface being of order a, whereas the element 

 of area of the cross-section is of order a^, it follows that the integrated transitory 

 displacements are of order a, and the integrated transitory stresses of order 0, that is, 

 of the same order as the applied tractions. 



As for the displacements and stresses arising from integration of the permanent 

 terms of the source displacements and stresses, their order is lower by one at least than 

 that of the corresponding quantities obtained by integration of the corresponding 

 transitory displacements and stresses. 



These results hold even when the applied force is discontinuous, provided it is 

 finite. When the applied force is continuous in z, and has a continuous z-derivative, 

 in the neighbourhood of a given section, it will be shown 



that, considering body and surface force separately, and considering force in any 

 one direction, any integrated transitory displacement or stress is of an 

 order higher by two than the corresponding integrated permanent 

 displacement or stress. 



In this case, then, the integrated permanent terras give results which are correct 

 as far as they go, and the error is of order al 



36. Modification of the preceding results ivhen the applied force, considered as a 

 function of z, involves the infinitesimal radius of the cylinder. 



The reasoning of last Article is based on the two hypotheses that 

 (i) the coordinate z is of order zero in a ; 



