3G 
MR. K. TERAZAWA ON PERIODIC DISTURBANCE OF LEVEL 
lias been obtained by Lame and Clapeyron* and by J. Boussinesq! ; the former 
by making use of Fourier’s theorem, and the latter by introducing several kinds 
of potential function. The solution for the case where the boundary condition is a 
normal pressure distributed symmetrically round a point on the surface and the body 
is free from bodily force is given by Prof. H. Lamb| in a form of definite integral. 
I have tried to solve the problem, independently of Lamb, and obtained the solution 
answering to any distribution of a normal pressure, from which the solution of Lamb 
can be derived as a special case. § 
For the present purpose let us confine our attention to the simplest case in which 
the boundary is subjected to a symmetrically distributed normal pressure. Take the 
centre of the area, in undisturbed state, on which the given normal pressure is 
distributed, as the origin of the co-ordinates, the inward normal to the undisturbed 
surface as the 2 -axis, and denote the distance of any point from the 2 -axis by r. 
Then corresponding to the normal pressure 
22 
=/M, 
(i) 
given at the surface 2 = 0, we shall have the following expressions for the displace- 
* 
ment components :— 
-f Z(k) 
/ / 
+ 
jfJ. J 0 
1 
2 (AT/x). 
e-HJj (hr) dk 
Z(k)e~ k *2 x (kr) 
u, — 
in which Z ( k) stands for 
f Z (h) e~ k *J 0 (hr) dk 
ZjUL J 0 
At 2 /* . 
ZyU \ A-1-yUjJ0 fi 
Z (/,•) = k f(x)J 0 (kx)xdx-, 
(3) 
(4) 
A, /u being Lame’s elastic constants, J 0 (x), J x ( x) Bessel’s functions of zeroth and 
first order, and a the radius of loaded area. 
Now, suppose the normal pressure (l) to arise entirely from the tidal loading in 
the North Atlantic basin, then the tidal loading would come into play to affect the 
water-level measurement at Chicago in two accounts : the one is the direct attraction 
of the material load, the other the deformation of the ground by the pressure 
* ‘Crelle’s Journal,’ vol. 7, p. 400 (1831). 
t ‘ Application des Potentiels.,’ Paris (1885). 
\ ‘ Lond. Math. Soc. Proc.,’ vol. 34, p. 276 (1902). 
§ The solution, with numerous examples, will be published shortly elsewhere. 
