170 MR JOHN DOUGALL ON AN ANALYTICAL THEORY OF 
There being no discontinuity of displacement or strain at the plane z=2’, except at 
the point where the force is applied, we are prepared to find that (iv) give no displace- 
ment at all if R>0. So long, however, as we keep to the specification of the strain 
by the , @ functions, it is convenient to retain the terms in (iv). By so doing, we of 
course make the @ of the space above the plane z=2' and the ¢ of the space below that 
plane two distinct potential functions, but we preserve the non-singular character of 
each of these functions at the axis R=0. 
If we take the limit of the above solution for z ie which obviously may be done 
by putting 2’ =h in each term and using the lower signs in (iv), we obtain simply the 
solution of (20), (21) multiplied by $(@+1). Since the present solution is for a force 
of 47u(a+1) units, and the other for an element of traction of 87» units, 1t follows that 
a unit element of traction may be regarded as simply the limiting case of a unit force, 
the point of application of which approaches indefinitely near the surface. 
21. Solution of a special problem of internal areal normal force. 
When the displacements due to a unit Z force at (’, y’, 2’), with the surface free, 
are known, the corresponding displacements for a body distribution of force, of 
amount Z(x’, y’, 2’) per unit volume at (a’, y’,z’), can be found by multiplying by 
L(x’, y’, 2)da’ dy' dz and integrating through the space in which Z is finite. Certain 
peculiarities in the form of the solution given in § 20 make it convenient to take the 
integration with respect to z’ last, or, as comes to the same thing, to ae by con- 
sidering the solution for an areal distribution of force on the plane z=2’, of magnitude 
Z(x’, y’, 2’) per unit area. : 
We take first a special problem analogous to that worked out in § 10, and suppose 
the Z force to be distributed over the area of a circle of radius a in the plane z=2’, with 
centre on Oz, the intensity per unit area being 47¢(a+1)J,,8ocosmm. It will be 
sufficient to attend to the value of ¢, for when that is known, the corresponding value 
of 6 can be written down at once. 
The series deduced by integration from (63), (64), say $2, fall naturally into two 
parts as in § 10, viz., (i) series defining a local hee at the cylinder p =a, 
sinh xz ne Seerl ; 
$= - eos) Qch = 1) { KZ sinh xz — get cosh 2«h)cosh xz f a Rr are (where sinh 2«h — 2«h=0) 
cosh Kz / ae rl. Qa 
fs Sele 2’ sie, ee xh — 2 = 
= ice eT 1 Kz cosh KZ! + 9 (cosh 2«h — a)sinh xz (ops at > (where sinh 2«h+2xh=0) 
with 
P, = J, Kp cos mo(KaG,,'KaJ ,, a — G,,xaBad,,’Ba), if p<a 
= G,, xp cos mu(kaJ,, Kad ,a—J,,Kka» Bad, Ba), if p>a 
(66) 
(ii) When p<a, series which can be summed in finite terms, 
$ = 27), 8 pcosma ey (| Be) saat ach 1) { Kz sinh KZ’ — : (a + cosh 2xh)cosh xz’ i (where sinh 2«h — 2xh = 0) 
F 2— K?)Kh(cosh 2«Kh — 
f Pa SHGaE TEED | Kz’ cosh kz’ + : (cosh 2«h — a)sinh xz’ i (where sinh 2«h + 2«h =0) 
7 — k*)«kh(cosh 2« 4 
(Gi) 
