CAUCHY ON-YHE THEORY OF LIGHT. 267 
sensible for points situated at very small distances from a certain 
plane drawn through the origin of the coordinates, and depend 
solely upon these distances. This same variable, and these de- 
rivatives, will not be sensible at the end of the time ¢, except in 
the vicinity of one of the parallel planes, constructed by means 
of the rule which we have before indicated. Consequently, if 
the sonorous, luminous, &c. vibrations are originally included 
in a plane wave, this wave, which we will call elementary, will 
divide into several others, each of which will be propagated in 
space, remaining parallel to itself, with a constant velocity. But 
these different waves will have different velocities of propaga- 
tion. If now we suppose that at the first instant several ele- 
mentary waves are included in different planes drawn through 
the origin of the coordinates, but little inclined upon each other, 
and that the sonorous, luminous, &c. vibrations are small enough 
to remain insensible in each elementary wave taken separately ; 
then these vibrations not being capable of becoming sensible 
_ but by the superposition of a great number of elementary waves, 
it is clear that the phenomena relating to the propagation of 
sound, of light, &c., can be observed at the first moment only 
within a very small space around the origin of the coordinates, 
and at the end of the time ¢, only in the vicinity of the different 
sheets of the surface which will be touched by all the elementary 
waves. Now, this last surface will be precisely the curved sur- 
face of which we have spoken above, and which we generally 
name “the wave-surface.” 
This being established, if we consider the motion of propa- 
gation of the plane waves, in a system of molecules acted upon 
by mutually attracting and repelling forces, we may take succes- 
sively for principal variables three rectangular displacements of 
a molecule m, measured parallel to the three axes of a certain 
ellipsoid, which will have for its centre the origin of the coordi- 
nates, and will be easy to construct as soon as we know the co- 
efficients depending on the nature of the proposed system, and 
the direction of the plane A BC, which included a plane wave at 
the first instant. Then this wave will divide into six others, 
which will constantly be of the same breadth as the first, and 
will be propagated with constant velocities in planes parallel to 
ABC. These waves, taken two by two, will have equal velo- 
cities of propagation, but in contrary directions. Moreover, these 
velocities, measured in the direction of a right line perpendicular 
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