912 DR JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



In terms of displacements 



Mp = { (z- y')"- + ~^r \ cos (oj - 0)'), 



1 2 



?<j = - 2(z - z')p cos (o) — o)'), 



with the rigid body translation 



Mp = COS (o) - oj'), ) 



Wu, = - Sill (oJ - 0) ), ^ -I 1 I 



„^^0 '1 ■«l27r;. 3,rM8-v)2 | 



(17) 



;i8) 



13. Permanent termfi for unit element of surface traction. 1)1 = 0* ?S^z', 

 Normal traction. 9(1). 



'^-~«2";rM8^)^'"^-^' ^="r7 4^;M8^.)^'~'^' ■ • • 



or 



1 v-4 



a 47r/A(8 - v) ' 

 Transverse traction. 9 (iZ). 



or 



11, ,, 



rt** TTfJ. 



Longitudinal traction. 9(3). 



, 1 1 



^42.;(^-^{^(^-^')^-vl-^=o 



where Ao and Bq are constants which need not be calculated ; or 



u^ = z~z'. 



1\ 1 



^- -iV-,,- 



aV 'i-/a(8 - v) 



(1) 

 (2) 



(3) 

 (4) 



(5) 



(6) 



14. Resolution of the displacement in an infinite solid due to a single force into 

 its 4>, ^, ■^ components. 



As a preliminary to the investigation of a solution for a single force acting at a 

 given internal point with the cylindrical surface free, it is convenient to express 

 the fundamental solutions corresponding to a single force in an infinite solid in 

 the form 2 (7). 



* Tiie expansion of D^ begins witli tlie term ^(v - 8)^\ 



