974 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



In (24), (31), (37) we have already found the equations for W_2, n_2 ; W_i, O.i ; 

 and U_2, V_2, Wq, Oq. In next Article a simple method will be given for determining 

 all the unknown functions of 2 which appear in (17), (21), (27), and (34). 



49. Determination of the unknown functions of z which appear in the asymptotic 

 solution. 



We have seen in these previous Articles that when the body forces are of order zero, 

 and the cylindrical surface tractions of order 1, the leading terms of an asymptotic 

 particular solution for these forces is given by Art. 48 (1), (17), (21), (27), (34). 

 The complete asymptotic solution differs from this by displacements of the type set 

 down in Art. 26 ; these may be of any order, so long as the conditions at the ends are 

 unspecified. The complete analytical solution of the problem involves besides certain 

 transitory perturbations which the method of analysis of last Article is incompetent 

 to determine. 



But even although the form of these transitory perturbations is unknown, we may 

 fairly assume, what in the earlier part of the paper was rigorously proved, that they 

 fulfil the relations of Art. 45. In fact, the method of Art. 46 shows that, for a 

 deformation of the cylinder under no forces in the body or at the cylindrical surface, 

 the value of each of the integrals in Art. 45 is a rational integral function of z of degree 

 not higher than the third ; and for any purely local perturbation this value is obviously 

 very small at a section remote from the region of disturbance. 



Hence the value of each of the integrals of Art. 45, as calculated from the 

 asymptotic solution, is exact. And it was shown in Art. 46 how the integrals can be 

 expressed in terms of the applied forces and of the tractions and displacements at any 

 assigned section. One result was written down explicitly. It may be put in the form 



(IV) = l7rEa4(F + F_2 + F_, + Fo + Fi), (1) 



where -T-f = 0, and the other functions are those defined in Art. 38. We will now 



calculate the value of the integral (IV) from the values of u, v, w and their derivatives 

 as given by (17), (21), (27), (34) of Art. 48. The only terms of these which 

 contribute to the result are easily seen to be those belonging to the 2a:;-flexural type 

 of deformation. We find 



(lV) = l7rEa2(U_2 + aU_i + a2Uo + a3Ui), (2) 



Comparing (1) and (2), we see that a"^U_2 , a"^U_i, Uq and aUi differ from F.2, 

 F..1, Fq, and Fi respectively by functions of the type F, that is, rational integral 

 functions of z only, of the third degree. 



The various functions V, W, O of Art. 48 can be found in the same way. The 

 functions thus obtained only diSer from those defined in Art. 38 by functions defining 

 permanent free modes as in Art. 26. 



