944 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



suppose the problem solved for a force F at (^', »?', ^') in the cylinder ^^ + »?'^= 1, so that 

 the displacement in a certain direction is (F/m)/(^, f], t, ; ^\ n' , V,)^ then the correspond- 

 ing displacement in our problem is (^\ixa)f{x\a, yja, z/a ; x'/'a, y'a, z'/a), for this 

 satisfies the differential equations of the problem in the body and at the surface, and 

 takes an infinity of the proper form at {x', y', z'). 



Let us now put p = ar, p = ar , where r and r' are considered to be of order zero in 

 a ; in other words, let us consider p and p to be, like a, infinitesimals of the first order. 

 But let z and z' be considered to be independent of a, i.e. of order zero. 



Under these suppositions, the parts of the displacements and stresses arising from 



the permanent terms of the single force solutions behave quite differently in respect of 



their order from the parts which arise from the transitory terms. 



JT {z — z'Y ... 

 For such a term as — ^^ '— in a displacement is of order — (n+ l), while a term 



ij.a dP- 



of the form — e " is of order — 1 when z = 2', and when z>0' may be said to be of 



infinite order. 



F iz — zY F '^'^-^) 

 Similarly with typical terms for a stress component, say — -^ ^ and — e "• . 



It has to be noticed that the /3-series for the displacements and stresses do not all 

 converge when z = z'. But we know that the displacements and stresses at = 2' are 

 actually finite except at the single point where the source is situated. Hence the 

 /S-series must tend to finite limits as z tends to z', except at the source, and the order 

 in a of the corresponding displacements and stresses can be rigorously inferred. For if 

 y(/3) is of order in a, then to find 



we may put z — z' = ae, and the limit becomes 



Lt2,e-^!/(/3), 



6 = ^ 



which is explicitly independent of a, i.e. of order 0. 



It follows that at the 2-plane on which the source lies, the displacements and stresses 

 from the transitory terms are of the same order as those from such of the permanent 

 terms as do not contain z — z'; and a term involving the factor {z — z'Y in a displace- 

 ment or stress is of an order lower by n than the part of the displacement or stress 

 arising from the transitory terms. 



35. Order of magnitude of the displacements and stresses in the solutions for body 

 and surface distributions of force. Comparison of the component parts of the 

 solutions arising from the jyermanent and transitory terms respectively in the source 

 solutions. 



When we come to solutions for extended distributions of force, which are derived 

 from the source solutions by integration as to the source coordinates, the distinction 



