Partial Differential Equation of Vibrating Plate. 
5 
parent, and their physical significance, should there be any, are more 
likely to be understood. Mere analytical or formal analogies may be of 
some help, and it is only to two or three of these that attention is called 
in this paper. The first is that of an elastic bent circular plate, and it is 
compared with an iron core under magnetic force. 
It is shown in treatises on elasticity and mathematical physics that in 
the “ case of a uniform plate of finite or infinite extent, symmetrically 
influenced in concentric circles by a load distributed symmetrically 
and by boundary appliances if required,” if r and 0 be polar coordin¬ 
ates of any point P, so that x—r cos 6 and y—r sin 0, the origin being at 
the center of symmetry, then the usual differential equation for the 
bending of an elastic plate having equal flexibility in all directions, viz.: 
(/).... A 
d 4 z Q c Vz 
dz 4 ~^'dx' 2 dy 2 
, d 4 z\ 
+ 7f?)= Z - 
dM 
dx 
dj X \,lx‘ + df/ Z 
takes the form 
( 2 )... 
A _d_( d_ 
‘ r dr ( T dr 
* Derived from the more general equation 
. d 4 z . d 4 z . rsy n \ d 4 z d A z _,d 4 z „ dM dL 
A dx i+ ~ b EjcW; + ( C ^ 2c ^dxMy i + ~ a dxdf +B dff 4 ~ Z ~dx^fdty 
(where A, 5, (7, c B, a, are supposed constant and not as in the most gen¬ 
eral case functions of x and y) on the supposition that all the coeffi¬ 
cients of elasticity are equal to any one of them, say A. As to the 
meaning of the constant A it is 
{l + d)pr 
-r 
In this expression 6 is Poisson’s ratio -= — — — ratio of lateral contrac¬ 
tion to longitudinal extension, which is also 
_ 2{2k + n) m—n 
3 k — 2n 2 m 
where k is the bulk modulus and n is the rigidity. 6 according to 
Poisson is always equal ^.t In very many known solids k is greater than 
f n, while in others it is less, so that there seems to be no necessary 
relation between these two constants. 
where q is Young’s modulus 
qr 6 
12{1—G) 
Skn _ 9kn 
m 3k + n 
and r is the thickness of the plate, and p its density. 
t Poisson and Navier’s assumption that this ratio was always = i 
k 
leads to the constant ratio — = f. Poisson always maintained that the 
