268 CAUCHY ON THE THEORY OF LIGHT. 
to the plane A B C, for the three waves which move in the same 
direction, will be constant, and respectively equal to the quo- 
tients which we obtain by dividing unity by the three semi- 
axes of the ellipsoid above mentioned. The points situated with- 
out these waves will be in repose; and if the three semi-axes of 
the ellipsoid are unequal, the absolute displacement, and the ab- 
solute velocity of the molecules, in a plane wave, will always re- 
main parallel to that of the three axes of the ellipsoid, which 
will be reciprocally proportional to the velocity of propagation 
of this wave. But if two or three of the axes of the ellipsoid 
become equal, the plane waves, which will propagate themselves 
in the same direction with velocities reciprocally proportional to 
these axes, will coincide, and the absolute velocity of each mole- 
cule included in a plane wave will be, at the end of any given 
time, parallel to the right lines along which the initial velocities 
were projected on the plane drawn through the two equal axes 
of the ellipsoid, or even, if the ellipsoid is changed into a sphere, 
in the direction of these initial velocities. 
Now, let us suppose that at the first instant several plane 
waves, slightly inclined one upon another, and on a certain plane 
ABC, meet, and are superposed at a certain point A. As the 
time increases, each of these waves will propagate itself in space, 
giving rise, on each side of the plane which originally included 
it, to three similar waves included in parallel planes, but pos- 
sessed of different velocities of propagation; consequently, the 
system of plane waves, which we at first considered, will be sub- 
divided into three other systems, and the point of meeting* [point 
de rencontre] of the waves, which will make a part of one and 
the same system, will be displaced in the direction of a certain 
right line, with a velocity of propagation distinct from that of the 
plane waves. Thus, at the end of any given time ¢, the point A 
will be succeeded by three other points, whose positions in space 
may be calculated for a given direction of the plane A BC, and 
the different positions which the three points in question shall 
take for different directions originally attributed to the plane 
ABC, will determine a curved surface of three sheets, in which 
each sheet will be constantly touched by the plane waves, which 
will make part of the same system. Now, this curved surface will 
be precisely that of which we have already spoken above, and 
which we have named “the wave-surface.” 
Finally, in order that the propagation of the plane waves may — 
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