crystalline Reflexion and Refraction. 19 
parallel directions; and therefore if we conceive a closed surface of any form, 
including any volume great or small, to be described in the quiescent ether, and 
then all its points to partake of the motion imparted by the waves, any slice cut 
out of that volume, by a pair of planes parallel to the wave-plane and indefinitely 
near each other, can have nothing but its thickness altered by the displacements; 
and since the assumed preservation of density requires that the volume of the slice 
should not be altered, nor consequently its thickness, it follows that the displace- 
ments must be in the plane of the slice, that is to say, they must be parallel to 
the wave-plane. And conversely, when this condition is fulfilled, it is obvious 
that the entire volume, bounded by the arbitrary surface above described, will 
remain constant during the motion, while the surface itself will always contain 
within it the very same ethereal particles which it enclosed in the state of rest ; 
and all this will be accurately true, no matter how great may be the magnitude 
of the displacements. 
Let x, y, z be the rectangular coordinates of a particle before it is disturbed, 
and a+ y+ 4, 2+ ¢ its coordinates at the time ¢, the displacements & y, ¢ 
being functions of x, y, z and ¢. Let the ethereal density, which is the same in 
all media, be regarded as unity, so that drdydz may, at any instant, represent 
indifferently either the element of volume or of mass. Then the equation of 
motion will be of the form 
SS dedydz papell a¢)= (SS dedydztV, (1) 
t dt dt 
where V is some function depending on the mutual actions of the particles. The 
integrals are to be extended over the whole volume of the vibrating medium, or 
over all the media, if there be more than one. 
Setting out from this equation, which is the general formula of dynamics 
applied to the case that we are considering, we perceive that our chief difficulty 
will consist in the right determination of the function V; for if that function were 
known, little more would be necessary, in order to arrive at all the laws which 
we are in search of, than to follow the rules of analytical mechanics, as they have 
been given by Lagrange. The determination of V will, of course, depend on the 
assumptions above stated respecting the nature of the ethereal vibrations; but, 
before we proceed further, it seems advisable to introduce certain lemmas, for the 
purpose of abridging this and the subsequent investigations. 
pd 2 
