272 CAUCHY ON THE THEORY OF LIGHT. 
having but one optical axis from the first formule which I had 
given. But the new analysis which I have employed leaves no- 
thing to desire in this respect, and extends to all possible cases. 
I shall also show in the second part of the memoir, that the 
pressure is evanescent in the ethereal fluid which propagates the 
luminous vibrations; and I shali show the conditions which the 
coefficients, contained in the differential equations of the motion 
of elastic bodies, should satisfy, in order that the surface of the 
luminous wave should acquire the form indicated by experiment. 
Lastly, in a third part, I shall show how we may establish the 
laws of reflection and of refraction at the first or at the second 
surface of a transparent body, and determine the proportion of 
light reflected or refracted. Here again, theory perfectly agrees 
with observation, and analysis leads me to the laws which seve- 
ral physicists have deduced from experiment. Thus, in par- 
ticular, calculation furnishes me with the law of Sir D. Brewster 
on the angle of complete polarization by reflection, and the law 
of M. Arago on the quantity of light reflected at the first and at 
the second surface of a transparent medium. I also obtain the 
formulz which Fresnel has inserted in the Seventeenth Num- 
ber of the Annales de Physique et de Chimie, and which would 
alone suffice to prove the truly wonderful sagacity of this illus- 
trious philosopher. 
Lastly, I shall investigate the means, by aid of which physi- 
cists may verify the reality of the triple refraction, or, what 
comes to the same thing, the existence of the third polarized ray, 
traversing a medium whose elasticity is not the same in every 
direction. 
Part THE SECOND. 
[Presented to the Academy the 14th of June, [830.] 
As we have seen in the first part of this Memoir, the integra- 
tion of the partial differential equations, which I have given in 
the ‘ Exercices,’ to represent the motion of a system of mole- 
cules acted upon by forces of mutual attraction or repulsion, 
leads directly to the explanation of the different phenomena — 
which the theory of light presents. But further, to establish — 
this theory, it is not necessary to have recourse to the general ~ 
integrals of the equations in question. It will suffice to discuss 
the particular integrals which express the motion of propagation — 
