46 
ME. K. TERAZAWA ON PERIODIC DISTURBANCE OF LEVEL 
Nevertheless the above analysis shows that the tidal effect on the water-level 
measurement, even at a point as far from the coast as Chicago, plays an important 
role and cannot be regarded as a small correction. 
IV. Elliptic Loading. 
Next, let us suppose that the tide in the North Atlantic is not uniform, but its 
surface is given by the equation 
2 2 
p + - 2 =l, z<0, 
b a 
viz., the excess pressure on the bottom due to the tide diminishes on approaching the 
coast so as to amount to 
zz = — (« 2 —r 2 )* for r < a , 
a 
r > a. 
( 26 ) 
In this case the function Z (k) becomes 
rr / 7 \ 7 [sin ka — ka cos ka ) 
Z[k) = — abgp \ --, 
( 27 ) 
Therefore we have 
u r = 
abg P z f [sin ka-ka cos ka\ T /7 _ N , 77 , 
Jo 6 I ^ } dL 
+ 
a 2 bg P (X + 2/j) |’ x [sin ka-ka cos ka) T /7 „ AJ7 , 
2n(\ + n) l 1 kV J ’ ’ 
aid 
7 3 
C\ 
ou 1 \ _ _ abgp (\ + 2p) 
9 
, cV /o Z p (A T p.) 
e -kz 1 
0 1 
12 2 
k a 
j 
z = 0 
( 28 ) 
• (29) 
The integrals contained in the above can be obtained by expanding the trigono¬ 
metrical functions into power series of k and making use of the formula 
Thus 
\ (hr) dk = k±T™± L IV 
Jo (r 2 +z 2 f? W(^+z 2 )J 
abgpz 
Ll z / 2 , 
p {r +z 
_ y / ■, p-i n{2n-l)\ [ a 
JnZL L) (2n+l)\ W(r* + z s ) 
\2n-l 
a J bgp(\ + 2p) “ , 1 Wl n (2n-2 ) l ( _ a X2n 2 
+ / u(X + / u)(?* s + 2 8 ) i n?/ ^ (2w+1)! \x/{r 2 + z 2 ) 
^2n-2 ( J/ )> • ( 30 ) 
