EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 921 

 and, for m>l, 



Af, =-2(m+l), A^^ =-1- ^-" , 



E^)„^ ^ - 2(.. + !)(« + .), B^;;„^ ^ - (,« - 1 + .) + ^|zz)L , 



C= 2(^+l)(- + v), Cj^,^= ^^^ + ('-l+v)-i|fli 



(15) 



All through this article, the forms given belong to the region p>p\ z>z'. 



For p<p', interchange p and p ; this changes one terra only in each of the nine 

 groups, that term, namely, which becomes infinite for p = 0, the rest of the terms being 

 symmetrical in p and p.* 



For z<z', interchange z and z ; this changes the signs of the whole of the terms, as 

 they all contain odd powers only of 2 — z'. In every case, therefore, (p, 6, and ^ are 

 even functions of z — z', as we may see that they must be from the definite integral 

 forms similar to 16 (9). 



It may be noticed, as a simple verification of the results of this Article, that in 

 each of the nine groups the source terms are balanced (Art. 15) by the others, that is 

 to say, each group of terms gives null traction at /) = a. But it must be observed that 

 this verification is not a complete one, owing to the existence of rational integral 

 values of (p, 6, \|^, giving no stress at the surface. None of the latter, however, involve 



a higher power of 2 than z^, so that, for a given source, — ^(<^, 6, ^) are determinate. 



If then we take, for example, the second part of 16 (11) and consider, in the source 



term, the coefticient not of -^ but of /3^, we can find particular values of (/>, 6, and -v/a 



to balance this and take a^^ of these particular values ; these will give the coefficients 



1 z-z' 



of—, seeing that any function / involving z only through the factor e~^~^ possesses the 



property ^, = -,f- 



For 7v'>[, it would be sufficient to consider in the source terms the coefficients not 

 of /8^ but of /3, for the only values of (f), 9, and ^^ which give no surface traction at jO = a 

 in this case are those which give no displacement at all (Art. 11). 



18. Solutions for a force at an internal point. P7'eliminary remarks. 



We have seen in Art. 14 how the <p, 0, and \|/ of the deformation due to a single 

 force in an infinite solid can be expressed by means of derivatives of {<p, 6, ^) =^T)^'^r~'^ 

 with respect to the coordinates of the point of application of the force. When the 

 balanced solutions which we have now investigated for the sources {(p, 0, \|/) = D7V"^ 

 (end of Art. 16) are subjected to the same differentiations, we get of course the same 



* When we come to calculate displacements due to a given force, the terms which vary according as p-> or < p' 

 disappear. 



