38 
ME. K. TEEAZAWA ON PEEIODIC D1STUEBANCE OF LEVEL 
III. Uniform Loading. 
Let a be the radius of the loaded circle, li the height of the material loading, which 
is supposed to be uniform, and p its density, then we have to put 
On this supposition, we get 
therefore 
u, — 
2 f± 
f(r) = —ghp for r < a, ] 
= 0 „ v > a. J 
Z (/■) = —ahgpJ 1 {ha), 
I e~ ,c *J 0 {hr) (ha) dh 
Jo 
+ ah 9hp+ 2 v) r e -kz Jo <u 
2p[X + p) Jo h 
( 8 ) 
(9) 
f du z \ __ __ ahgp (\ + 2^c) 
\ 07 * /o 2 JUL (X “("ybt) 
e-**^ (hr) J, (ifca) dk 
L 
• ■ ( 10 ) 
The other components of the displacement and those of stress can be expressed in 
similar forms. But it is unnecessary to write them down here as they are out of our 
present purpose. 
The integrals required here cannot be evaluated in a very simple way. Some of 
them are closefy connected to the magnetic potential due to a circular current, or to 
the velocity-potential and stream function of a circular vortex, and have been 
discussed by various authors. In his paper on the inductance of circular coils,^ 
Prof. Nagaoka has devised a comparatively simple method which may be applied to 
evaluate all the integrals needed for the calculation of the displacement and stress in 
the present problems. Let us follow his method and describe it here briefly. 
Put 
B = f{a/ + r~ — 2ar cos 0), 
then by Neumann’s addition theorem for Bessel’s function we have 
J, (hr)f {ha) = - f J, (kR) cos 6 d0, 
7T Jo 
Ju {hr )Jj {ha) 
i r 
7r . o 
a—r cos 0 
R 
Jj (/vR) do. 
* ‘ Phil. Mag.,’ VI., 6 (1903), p. 19. 
