214 REPORT—1885. 
forces (2) depend on U and its differential coefficients, and it is assumed 
in the theory that in consequence of the excessive smallness of U they 
may be neglected. Again, let us suppose that the dimensious of the ele- 
ment of volume are large compared with the distance through which the 
action of an ether particle on a matter particle is appreciable. Then we 
may consider the mutual reaction between matter and ether as confined 
entirely to the element of volume considered, the actions taking place 
across the surfaces of the element will just balance each other, and hence,. 
if we consider the matter and ether as one system, the force (2) will be 
zero, and the equations of motion will be 
du a?U dé 
mop +e ae = (A—-B) 7. + Bvu, ete. : Paka 
U is here the displacement of the matter occupying the same element of 
volume as the ether, whose displacement is u, but all the displacements. 
being very small, it is assumed that we may treat U and w as the dis- 
placements of the matter and ether, which when at rest occupy the same 
element of volume. Thus U, V, W are functions of wu, v, w and their diffe- 
rential coefficients with respect to 2, y, z, the initial co-ordinates, and may 
be expanded in terms of these, and it remains to find the form of the: 
expansion. 
Conditions are, of course, imposed by the fact that the medium is. 
isotropic, and it is shown that so far as second differential coefficients we 
may write 
dé 
U=Au+C7, +Dvu, etc. . : ae 
On substituting this value of U, in the equation of motion, and assuming; 
20 € mr seer) 
u=Me' = \'~ w etc., we obtain 
2 2 2 
(0-+p, A)ot= ( opti een ( Mc) e) vu. (8) 
dt" T dx 7 
And these equations, of course, give a normal wave travelling with a 
velocity [ {A + 2u + 4(C + D) x’p,/77} /(o + Ap,)]!, and a transverse 
wave with velocity [ {u +4Dz7*p,/7?}/(p + Ap,)]}. 
These velocities vary with the period of vibration in a manner which 
agrees, at least approximately, with experiment. It is clear that the 
coefficient A is positive, while the experimental fact that the velocity 
increases with the period shows that D is negative. The condition that 
A is positive merely implies that the ether tends to move the matter 
particles in the same direction as it moves in itself. 
If we suppose that the medium is not isotropically symmetrical, while 
at the same time it is such that the expressions retain the same form when 
two of the axes are turned through a small angle about the third, them 
terms B & -- =) come into the value for U, and these, it is shown, 
would cause the medium to produce rotation of the plane of polarisation 
of a plane polarised ray traversing it. This rotation would vary approxi- 
mately inversely as the square of the period, in accordance with the law 
discovered by Briot. By introducing higher differential coefficients into 
