CAUCHY ON THE THEORY OF LIGHT. 265 
The three partial differential equations which represent the 
motion of a system of molecules, acted upon by mutually attract- 
ing and repelling forces, include, with the time ¢, and the rec- 
tangular coordinates w, y, s of any point of space, the displace- 
ments £, 7, of the molecule m1, which coincides at the end of 
the time ¢, with the point in question; these displacements being 
measured parallel to the axes of the 2, y, z. The same equa- 
tions will offer one-and-twenty coefficients depending on the 
nature of the system, if we make abstraction of the coefficients 
which disappear when the masses m, m!, m'' of the different mo- 
lecules are two by two equal among themselves, and symmetri- 
cally distributed about the molecule mt upon right lines drawn 
through the point with which this molecule coincides. In short, 
these equations will be of the second order, that is to say, they 
will only contain derivatives of the second order from the prin- 
cipal variables £,7,¢; and, by considering each coefficient as a 
constant quantity, we may reduce their integration to that of 
an equation of the sixth order, which will include only a sin- 
gle principal variable. Now, this latter may easily be integrated 
by the aid of the general methods which I have given in the 
19th number of the Journal of the Ecole Polytechnique and in 
the memoir on the application of the residual calculus to ques- 
tions of mathematical physics. By applying these methods to 
the case where the elasticity of the system remains the same in 
every direction, and reducing the value of the principal variable 
to the simplest form, by means of a theorem established long 
ago by M. Poisson, we obtain precisely the integrals which that 
geometrician has given in the Memoirs of the Academy. But 
in the general case the principal variable being represented by 
a definite sextuple integral, it was necessary, in order to dis- 
cover the laws of the phenomena, to reduce this sextuple inte- 
gral to an integral of a lower order. This reduction stopped me 
along while; but I at last succeeded in effecting it, for the par- 
tial differential equation above mentioned, and indeed generally 
for all partial differential equations in which the different deriva- 
tives from the principal variable, compared with the independent 
variables xv, y, z, t, are derivatives of the same order. I then 
obtained, to represent the principal variable, a definite quadru- 
ple integral, and I was enabled to investigate the laws of the 
phzenomeaa, the knowledge of which should result from the in- 
tegration of the proposed equations. This inquiry was the ob- 
VOL. III. PART Xx. ir 
