452 The Rev. S. Earnshaw on a new Theoretical 
molecular action. This is the system of medial action supposed 
in the investigation which I have to produce. 
2. It is to be noticed, also, that when the medium is supposed 
to be continuous, the whole force exerted on F on one side is 
supposed to be exerted by the slice E, and on the other by the 
slice G; but on the other supposition, these forces are distr- 
buted among the slices E, D, C... on one side, and G, H, I... 
on the other, according to some rapidly decreasing law, which 
we shall have to determine. Consequently the force which on 
the first supposition is exerted by one slice (E) upon F, is on the 
latter hypothesis exerted by E, D, C... unitedly; so that the 
force in the former case exerted by E alone is equal to the sum 
of the forces exerted by E, D, C... in the second case. 
3. Let z be the distance of any slice D from F, and let H be 
at the same distance on the other side of F, Then we may repre- 
sent by mf(z) the force exerted by either of the slices D, H ona 
particle in F. Hitherto we have supposed the medium in equi- 
librium, let it now be in a state of motion; and for simplicity 
let us suppose all the particles in any slice to be in the same state 
of disturbance. Denote by 2,, 2, 2! the disturbances of the slices 
D, F, H at the time ¢. Then the whole force exerted by D and 
H on a particle of the slice F 
=mf(z+a—2,) —mf(z+a! —x) = —mf'(z) . {2,—22+2'}, 
neglecting powers of x—, and a!—wx above the first. Now 
this step supposes that the relative displacements of any two par- 
ticles which are within the sphere of each other’s action, is so 
small in comparison of their distance from each other, that the 
square and higher powers of it may be neglected. The absolute 
displacements may be of any magnitudes, subject to this con- 
dition. Our results will therefore not be limited to small abso- 
lute motions, but to small relative motions of particles within the 
sphere of each other’s action. Let now / be the thickness of 
the slices; and denote the disturbances of ...D, H, F, G, H... 
from their equilibrium positions by ...%~2, L-1, Ur» Veo iy Lppgoees 
respectively. Then the equation of motion of any particle of the 
slice F will be 
pe, mg? (h). (p=) — 2a, + By41) 
+ mf"(2h) . (€p-2— 2+ X49) 
+ mf" (3h) . (tp-3— 22, + Lp+3) 
4. It is not very difficult to exhibit symbolically the general 
integral of this equation; but that is no part of my present 
object, which is to find the velocity with which any disturbance 
is propagated through the medium. [I shall therefore, for the 
