ARISING FROM THE LOAD OF NEIGHBOURING OCEANIC TIDES. 
47 
where 
Vir’ + z 2 ) 
These series converge for x /(z 2 + r 2 ) > a, and are applicable in this region. 
At the surface we have to put z = 0 and v = 0 . Since 
we have 
C sin ha —ha cos ha T n \ m 1 -n /i jl 5 a ‘ 
Fa 3 
J 0 (hr) dk = i- F ( 1 h f, ^ 
Consequently 
3; 
JL_ 
2 a 
r 
l-f - 2 sin-^ + ^-fl- 
2 a 1 r 2a 
a 2 Y' 
Wc 
_ abgp (X + 2,u) 
4^ (X + ,u) 
r 
o * 
2 a - 
• -1« , r 
sm - + — 
r 2 a 
Ji-^1. . 
(31) 
(32) 
for r = a. 
To find the expression for the vertical displacement within the loaded circle we 
proceed as follows :— 
Making use of the power series of Bessel’s function, we have 
-kz 
sin ha —ha cos ha 
Pa 3 
J 0 (hr) dh = 
■- 1 '' ■ (33) 
where stands for 
a 
(*}= .1 
= I e 
0 
-kz 
a 71=0 (n !) 2 \ 2 a 
[sin X —X cos X i 
l 
X 3 
X ro d\. 
The evaluation of the function Q m ( x) can be undertaken by the aid of the formulae 
sm X 7 . , ,1 
e x - d\ = tan ] - > 
0 X x 
e~ Xx cos X d\ = 
x 
JO 
1 “I - cc 
2 
A little calculation will give us 
^0 ( x ) — ^ j x + tan 1 x—x 2 tan -1 — 
x 
Q 1 ( x ) = 1 — x tan 
Q 2 (x) = tan -1 — — 
_i 1 
x 
x 
X l+x“ 
,2 5 
(x) = 
(i +x 2 Y ’ 
