922 Dll JOHN DOUGALL ON AN ANALYTICAL THEORY OF THE 



singularity as before, but now no traction at the cylindrical surface ; that is to say, we 

 get a solution for the internal force subject to no surface traction. 

 Denote the values of (p, 6, \|/ in the balanced solutions by 



for the sources ^--DrV-i, (9 = I)^-V"^ «// = Dr^'-i , j ^' 



respectively, so that in the notation of (11), (13), (15) of Art. 16 



</'*=?>r' ■ • '• (2) 



and so on. 



Take now, for example, a Z force of iTrfxv units at (p', w', z'). 

 By 14 (17) the balanced solution for this is compounded of 



{cl,,e,xl.)=-p^-,^-,{i^e,e%xj.e),). (3) 



Pi 7\ 



I d . ,39, 98, 

 cz dp oz dm oz 



,9 ,9 9, 9 9, 



'^ = 9/'^-^97 9?^^"9-^'9^''^^' 



We may compactly represent the three balanced solutions of (l) by the symbols 

 S^, Sy, S,^ and the three balanced solutions for a U7iit P, ii, Z force by Sp, S^, Sz 

 respectively. We may then write, by 14 (17), 



i.e. it is defined by 



Sp = J_ i „-^S. + f vl - l,'lMe U'-.- IM.S 



Att/xv \ dp \ dp dp' dp' J \p' dp'Jdu) 



iTT/JLV ( p OCO \p op/ 0(1) \ Op p 0(1)-/ ) 



a _ 1 j3c;_'3 3c;_3 ^sl 



iirixv \ dz dp dz' 9a)' dz ) ' 



(5) 



the meaning being that in any one of (5) we may replace S throughout by *p or by 6 

 or by "^ ; so that the last equation of (5), for instance, is equivalent to the three 

 results of (4). 



Each of the solutions S^, S^, and S^ consists of a permanent and a transitory part. 

 These may be denoted by S^\ SJ^ ; 8(">, ^'■^ ; Sf, ^f ; and a similar notation may be 

 used with respect to Sp, S^, and Sj,,. 



The permanent parts of S^, S,,, and S^ are given in Art. 1 7 ; their transitory parts 

 are 2' of the first lines of (11), (13), (15) of Art. 16. 



The calculation of the permanent parts of the displacements in the solutions for a 

 single force by the method just indicated involves no difficulty, but it is somewhat 

 lengthy, and in the next article the final results only are given. The transitory terms 

 are left meanwhile in the forms (5). 



