Displacements of Circular Vibrating Plates. 
7 
of the plate which we may designate by Z. “ Z will then be the amount 
of load per unit area, when the applied forces on each small part are re¬ 
ducible to a single normal force through some point of it.” 
Small z is then a function of r likewise and denotes the displacement 
perpendicular to the original plane of the plate. Its value as is well 
known is: 
z — ~~f rdr§—f r Z dr + l C (log r — l)r- + i CV 2 + C" log r + C" 
C" is merely displacement of the plate as a whole and need not be 
any further considered. 
C" log r denotes anticlastic displacement with ±C"/r' 2 for the curva¬ 
ture, the same in the two principal sections. The radial and cross 
radial bending couples to match are 
L=±(A-C)f r . 
<7, the same. This would be realized in an infinite plane plate with a 
circular aperture and uniform distribution of load in the shape of bend¬ 
ing couple around the circular edge everywhere as axis. 
34 C'r 2 is the displacement of spherical curvature. 
34 C (log r—1) r- is a deflection involving shearing force and couple. 
■■SrAC/r— shearing force and %C(A + c) log r+A(A — C) = bending couple. 
Since from the symmetry of the case considered when the plate has 
a circular contour the tangent plane to the strained plate at the center 
will be horizontal, C" will be zero unless there be discontinuity in the 
“Circular loading so as to cause a circle of inflexion to occur between the 
center and the outer margin, which is without the limits of any cas e 
save a circular loaded membrane and not an elastic plate. It is realized, 
however, in the case of a plate with a circular aperture.* 
If we turn from elasticity to electro-magnetism and consider the in¬ 
duction of currents and magnetic lines in cores under the influence 
of coils wrapped about them and carrying currents, we shall eventually 
come upon an equation entirely similar to equation ( 2 ). For, as Oliver 
Heaviside has shown in discussing this subject of induction of currents 
44 a fluid analogous to the electric fluid ” having only a repulsion between 
its particles and having moreover V, the potential, only a function of 
the distances between these particles. But our medium would have to 
be one in which every particle would have both attraction and repulsion 
for its neighbors according to the poles presented, and ultimately a re¬ 
pulsion according to an unknown law to account for first the extension 
of a bar of soft iron under medium magnetic stress, as first shown by 
Joule and as measured by Mayer and others, and afterwards a contrac¬ 
tion, as Shelf ord Bid well-has shown. 
■ * Thoms on and Tait Nat.-Phil., §651. 
