900 



DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



The displacements in cylindrical coordinates can be written down at a glance from 

 (2), (3), and (4) ; they are 



d^ 

 ^^'W 



d-d> d<f> 

 opdz dz 



oz 



p 9(1 



dp' 



dW 



These give by (4) of Art. 1 * 



(U, dZ' opoz^ op^ 



pw_d^ 2_ d^6 2 8(9 



yu. 8(1)82''' p 8p3(o p- 8(1) 



px: _ ., 83^ _ 82^ 2. i!^ . 9^ 



p, dz'^ dpdz p dw'^dz dpdz 



p 8p8(o p2 8(j) 



_ 8V J_ 3^ 1 8V 



8p2 p 8p p2 8(0^ ' 



p 8(io8z 



(7) 



(8) 



The symbols cp, 0, and ^ will be reserved throughout for harmonic functions defining 

 a deformation of the cylinder as in (7). 



It will appear that any deformation whatever, within a part of the solid to which 

 no body force is applied, can be specified in this way. The specification is not unique, t 

 as may easily be seen, but this leads to no practical inconvenience. 



3. Solution of a sirrtple problem of surf ace traction. Fundamental free modes of 

 equilibrium, pe7'mane7it and transitory. 



We begin with a simple problem, the solution of which will be useful hereafter. 



Let the eiven surface tractions be 



/3Z 



where Pq, ^oj ^0 are constants. 

 Assume 



PP = P„e a cos ni((o - co'), 



_ -^^ . 



poi = fi^e n sin m{ii} - o) ), 



^ -^ 



p2 = ;SZj,e « COS m(o) - 0/), 



<h —■ AJ ,,,—e a cos ??2(co — (u'), 



a 



ftp _^? 

 6 = BJ,„^e a cos m(oj - oj ) 

 a 



Bp -^ . 

 xj/ = C J„,— e a sin ?w(oj — 0/), 



(2) 



or, in a compact notation which will frequently be used, 



V rv c r-i' «sin^(--''^) 



(3) 



* Such a reference will be given briefly in the form, 1 (4), 

 t Cf. Art. 11. 



