942 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



and, from 30 (5), 



\^= - 16 V27r"= COS N7re~T(N7r)5. . 



(4) 



The solution for a unit element of normal traction at (a, w', z) contains (Art. 6) 





C 



iiuNiT- 



ap 





^2tN37r3^^, 2N37r3'^ /, N 



2mW: 



,a-p 



o 



sin ^ ' I 



(5) 



For longitudinal traction multiply (5) by ( - i) ; "j 



for transverse traction multiply (5) by ^tj— , and change . into ' ?)i(w - w'). I " ' w 



For a unit force at an internal point (/)', w', z') multiply the results for the 

 corresponding surface force by the coefficients 



(P force), - 1 I ?!'JSM« - P) {^) -h''^'-i) \ , 



(O force), i-j . L I (7) 



(Z force), i{ . |, 



J 



where 



e '^V n / = (4N7r) ia e a . 



Remark that 



(i) the displacement u^ and the stresses fw, zw are relatively negligible, so that 



the mode is approximately non-torsional ; 

 (ii) the mode is less important for transverse than for normal or longitudinal 



force ; 

 (iii) for m = 0, it is exactly non-torsional, and does not occur for transverse force. 



33. Remarks on the appi'oximate forms of the transitory modes. 



It should be observed that the results of the two preceding Articles fail when p or 

 p' is zero. For these cases the values of the various quantities are of course easily 

 found independently. 



We verify at a glance that the principal terms of the p-tractions vanish at the 

 cylindrical boundary. Moreover, we see that in Art. 31 the displacement m^, and in 

 Art. 32 all the displacements, vanish to a first approximation at the boundary. 



It is interesting to verify the results of Art. 29 from the above asymptotic forms. 

 In the case of a mode associated with a complex zero of D,„ this involves the 

 independent calculation of the surface displacements (or at least of one of them) to a 

 higher approximation, but this can be done without difficulty. 



1 



