48 
MR. K. TERAZAWA ON PERIODIC DISTURBANCE OF LEVEL 
and, in general, 
a„(x) = (-1)*- 1 
d m ~ 3 f 2 \ 
C?£C m-3 V(l +x 2 )/ 
m > 2. 
Thus the integral on the left-hand side of (33) can be expanded in an ascending 
power series of r/a which probably converges for limited values of r if the value of 2 
is fixed. This series and that found in (30) have a common region in which they 
are both convergent and therefore they must be congruent to each other in that 
region. On the proof of this proposition we shall not enter, but we shall find the 
region of convergency of the latter series at the boundary. 
Expand Q 2n 
into a power series of z/a , supposing 
z/a to be sufficiently small, 
then the first term of it will be ( —1)' ! 2 (n— 1) (2n — 2)! — . If we retain only the 
a 
terms which contain the first power of z/a in the series of (33), its general term will 
then be 
2 (n-1) (2u —2) ! l'r\ 2n _ 2 
(n !) 2 2 2 " \aj a 
The series which has this expression as its general term converges obviously for 
the values of r smaller than a. Thus the expansion (33) applies for r <a when 2 is 
an infinitesimal. 
Since, for 2 = 0, 
Q„(0) = j, Q,(0) = |, Q 4 (0) = 0, Q, (0) = 0, 
we have 
for r ^ a. 
Hence 
sm ka kcc cos l\Cx t / 7 \ 7 7 i 
- — -J.(ir) *=-(!-, 
Cl 
J.( 34 ) 
( 35 ) 
for r = a. 
Quite similar arguments may be employed to find the expression for the tilting. 
We shall have 
sin ha — ka cos ha 
k?a? 
J, (kr) dk = -X 
t f • i a a 
sm - 
a 
r*V 
irT 
4a 2 
r r \ ] J 
r — a. 
• (36) 
