EQUILIBRIUM OF AN ISOTROPIC ELASTIC ROD OF CIRCULAR SECTION. 967 



certain definite sections, including the ends, supposing the radius of the cylinder to 

 be small. 



In the present Article we give an independent investigation of the form of the 

 chief terms in the solution for a finite cylinder. The nature of the analysis is such 

 that the trnnsitory terms necessarily escape detection, but for practical purposes this is 

 of comparatively little consequence. 



We assume that the value of each of the displacements in any solution is capable 

 of being arranged in a series of terms of ascending order in the radius a, the order 

 increasing by one from term to term. The series need not be convergent, but we 

 assume that similar series for the derivatives of the displacements of the first and 

 second orders can be found by term-wise differentiation of the series for the 

 displacements. 



We take x = a^, y = wi, so that the cylindrical boundary is ^^ + 'y^ = 1 . 



The body forces are a^X(^, »?, z), c»^Y(^, rj, z), a^Z{^, *], z), and the tractions at the 

 curved surface a^+^X.(^, r,, z), a''+%(^, >i, z), a^+'Z^^, r,, z). 



For convenience, the value of p will be chosen so that the lowest terms in the 

 displacements are of order — 2 ; and in the general case when the forces specified 

 above are connected by no relation, it will be shown that p is zero or positive. 



We shall write u, v, iv instead of u^, Uy, u^. 



Then . 





(1) 



where w,., v,., tv^ are functions of ^, /?, z. 

 The body equations of equilibrium are 



^d^u d'^u d^u\ ,, , ^d /du dv div\ , „„ r. 





dx\dx dij dz ) 





(2) 



and the equations at the curved surface 



X ( ,/9m dv dw\ , „ 9w I y fdu dv\ „„+i^ 



X /9m dv\ y ( ./du dv dw\ 3" I ^^jj+iy 



a \dy dxj a\ \dx dy dz) dy ) 



X fdu dto\ y fdv div\ ,,+i v 



a \dz dxJ a \dz dy/ 



(3) 



Now -- = -— , --„ — -„ -rr., and so on ; so that ( , — )Ur are of order — 1 . and 



dx a 94 dx^ a 34 \dx dy^ 



--^ \Ur of order — 2. 



dx dxdy' dy 

 TRANS. ROY. SOC. EDIN., VOL. XLIX. PART IV. (NO. 17). 



132 



