142 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
2. Force applied at a single point. 
Returning to the solutions (i), (ii), (111), we note that (i) and (11) contribute nothing 
to the dilatation A, and (ii), (iii) nothing to the z-rotation v= 5(E- a) . 
These properties can be used to resolve any given displacement into its , 0, > 
components, the bodily force being null. 
An example of fundamental importance is the displacement in an infinite solid due 
to a single force applied at a given point. ‘Thus for a unit force applied at the origin 
in the direction of the axis of z we have 
a) 
=e h multiplied by 1 At# 1 
re each multiplied by Sere eT) 
p= 2 Ae 
- Bor 
where 7° =x" +y’+2; or say, for a Z force of 47m (a+1) units applied at (w’, y’, 2’) we 
have 
(6) 
1 
w= (Z- = +a! 
r~* being written for 1/r, where + is the distance from (a, y, z) to (x’, y’, z’). 
These give 
ae oe a0 
But in (4) E 
A=2%1 -a)o@ oa 
Hence we take = 0, and choose ¢ so that oo - = or, 
Now the functions log (r +z—z’) and—log (r—z—Z) are both potentials having 7~? for — 
z-derivative ; the former is without singular point in the region z>z’, the latter in the 
region <2’. We may without confusion use a single symbol to denote either function 
indifferently, and define 
« 
dy 1 
log (r+2—2' Z>e2 
ap og(r+z—-z) when z2> | (1) 
= —log(r—-—z+2') when z<7 
We may therefore take 
_ ier 
t 2 ae 
