266 CAUCHY ON THE THEORY OF LIGHT. 
ject of the last memoir which I had the honour of presenting to 
the Academy, and which contains, amongst others, the following 
proposition. 
A partial differential equation being given, in which all the 
derivatives of the principal variable relative to the independent 
variables #, y, z, ¢, are of the same order, if the initial values of 
the principal variable and of its derivatives taken with relation 
to the time, are sensibly evanescent in all the points situated at 
a finite distance from the origin of the coordinates, this vari- 
able and its derivatives will no longer have sensible values at 
the end of the time ¢, in the interior of a certain surface, and 
consequently the vibrations, sonorous, luminous, or other, which 
may be determined by means of the partial differential equation, 
will be propagated in space, so as to produce a wave, sonorous, 
luminous, &c., whose surface will be precisely that which we 
have just indicated. Moreover, the equation of the surface of 
the wave will easily be obtained, by following the rule which I 
proceed to describe. 
Let us suppose that in the partial differential equation we 
substitute for any derivative whatever of the principal variable, 
taken in relation to the independent variables 2, y, z, ¢, the pro- 
duct of these variables raised to powers, the degrees of which 
are indicated for every independent variable by the number of 
differentiations relative to it. The new equation obtained will 
be of the form 
F (yseod) = 0, 
and will represent a certain curved surface. Now, consider the 
radius vector drawn from the origin to any point of this curved 
surface; take upon this radius vector, setting out from the 
origin, a length equal to the square of the time divided by this 
same radius; then draw through the extremity of this length a 
plane perpendicular to its direction. This plane will be the 
tangent plane to the surface of the wave, and consequently this 
surface will be the envelope of the space which the different 
planes will traverse that may be constructed by the operation 
just described. Finally, we come to the same conclusions by 
following another method, which I will now state in a few words, 
and which I explained in my last lectures at the College of 
France. 
Let us suppose that the initial values of the principal variable 
and of its derivatives taken with relation to the time, are only 
