EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 905 



No increase without limit while NuO-^, remains equal to unity. The limiting form of (6) 



is, by (8), 



(V0%i2X<H(V'>>''»"(-'-)r»("-"V(*'^f-')}. . . m 



We shall call (9) the solution for a unit element of normal traction at {a, «', z'). 



From (8) we see that the solution for any distribution of normal force over a 

 finite area can be derived from this by multiplying by N(ft)', z') and taking the surface 

 integral over the area with respect to the accented coordinates. 



Lastly, in the study of the solution, we may simplify matters still further by con- 

 fining our attention in the first place to the part of (9) depending upon a single integer 

 m. This is 



(V0-Ai/'"K%>'''""<-^'):£'"("-'">(*7'")}- • ■ <"' 



6. Unit element of normal traction. Transform^ation of the solution in definite 

 integrals into a solution in series, by Cauchy's Tlieoi'em. 



The m-component of the solution for a unit element of normal traction at (a, «', z') 

 is, by 5 (5), 5 (10), 



In each of the three integrals here, the integrand is a uniform, even function of k 



of the form 



E(k) cos k(z - z) - E„(k), 



where E„(k:) stands for the terms of negative degree in the expansion of E(/<:) cos k{z — z') 

 in ascending powers of k. 

 Now 



r{E(K)cosK(z-z')-E»W}'^'« = i/ {li('<)cosK(z-z')-E„(K)}(''K. ... (2) 



The latter integral may be regarded as a complex integral, and its path of 

 integration can be deformed. 



Two paths of integration in the ^-plane which will be used frequently are defined 

 as follows. Let a circle of radius e be described about the origin as centre, where e is 

 a positive quantity which for convenience will be taken to be small. Then the first 

 path passes along the real axis from - <» to -e, along the upper semicircle from 

 - e to e, thence along the real axis from e to oo ; the second 'path differs from this only 

 in taking the lower semicircle. 



Thus the right-hand member of (2) 



= 1 /"{E(k) cos k{z - z) - ^n{x)]dx, (first path) | 



r \ ^^^ 



= \ I E(ic) cos k{z - z')dK, (first path), , J 



ci K 



for evidently -^ over the first path vanishes. 



J 



