40 
MR. K. TERAZAWA ON PERIODIC DISTURBANCE OF LEVEL 
then we shall have 
r a—r cos 0 _ a r 2 —a 2 
Jo E 2 (R 2 + z7 “2 a Ml 4 a 2 r 
du 
9 (v) — 9 (u) 
R 2 
Now we have 
a—r cos 0/-p 2 , 2 u 2/ , \ r 2 — a 2 , r 2 — a 2 f“ 2 9(v) — e 1 
-(R +0- b<9 = — (e 1 o ) 1 + > /1 ) - - 5 —,®!+ 1 - L 
CTa 
arcC 
a 2 ra. 2 E, 9 (v) — 9 ( u) 
du. 
'“ 2 9' (v) du 
oj 3 
P (v)—9 (u) 
Hence 
77 a— r cos 6 a 
i aO = — 
loR 2 (R 2 + z 2 ) 
a—r cos 0 
log 17 j'' + ' | — 2w£ (v) 
S <r{u-v) V 7 _U 
= 2vt] 1 — 2w^ ('y) + 2m7rt. . . 
(vi?!—®if (v) + miri}, 
2 2 
r—a 
2a a 1 2a 2 r9' ( v) 
(15) 
(16) 
R 2 
(R 2 + z 2 ) 4 c £0 = — («!«! +>a)-■ 
r 2 —a 2 
act. 
2 2 
a ra 
, 2 (r 2 —a 2 ) 9 (v) — e 1 s \ , m 
(17) 
The term m-rri enters because of the many-valued property of a logarithm. The 
actual value of m and 9' ( v ) will be determined by the following consideration. 
From the definition of 9{v) and e u e 2 , e s , it follows immediately that 
9{v)-e i 
9 {v)~ e 2 
9 (v) — e 3 
Z 2 a 
2 ar 
(a —r) 2 a 
2 ar 
(a + r) 2 a 
2 ar 
accordingly 
e 2 <9 ( v ) < e 1# 
The last inequality shows that the value of v must be one of the following 
(i.) v = (2n+l) w 1 + {2n’ + 0) w s 
(ii.) v = (2n+1) « 1 + (2n' + 2 —0) o) 3 _ 
. (18) 
where n and n' denote any integers, positive or negative, or zero, and 6 a positive 
number less than unity. To determine the value of m in the formulae (16) (17) for 
