330 M. La Place’s Investigations on Double Refraction. 
incident and refracted ray are in the same plane, and that the 
ratio of the sines of their inclination to a vertical line is constant. 
In the next case, the action of the medium upon light is equal 
to a constant quantity, flus.^ a term proportional to the square 
of the cosine of the angle which the refracted ray forms with 
the axis ; for, as this action is equal on all sides of the axis, it 
must depend only on the even powers of the sine and co-sine of 
that angle. The expression of the square of the interior velo- 
city is thus of the same form as that of the action of the me- 
dium. By substituting this expression in the differential equa- 
tion of the principle of least action, M. de La Place then deter- 
mines the formulae of refraction in relation to this case, and he 
finds that they are identically those which are given by the law of 
Huygens. Hence it follows, that the Huygenian law satisfies 
both the principle of least action, and the condition that the in- 
terior velocity depends only on the angle formed by the axis and 
the refracted ray. 
M. de La Place then proceeds to remark, that the hypothesis of 
Huygens, that the velocity of the ray is expressed by the va- 
riable radius of the ellipsoid, does not satisfy the principle of 
least action, but that it satisfies the principle of Fermat, which 
consists in this, that the light arrives from a point taken without 
the crystal, to a point taken within it, in the least time possible. 
For ,it is obvious that this principle becomes the same as that of 
least action, by reversing the expression of the velocity. Hence, 
both these principles conduct to the law of refraction discovered 
by Huygens, provided that, in the principle of Fermat, we as- 
sume with Huygens the radius of the ellipsoid as a measure of 
the velocity, and that in the principle of least action, we as- 
sume this radius as representing the time employed by light in 
traversing a determinate space taken for unity. 
The identity of the law of Huygens and the principle of Fer- 
mat results, as M. de La Place has remarked, from the ingeni- 
ous way in which Huygens considers the propagation of the 
waves of light, so that his way of considering it, though very 
hypothetical, represents nevertheless all the laws of refraction 
which may be due to attractive and repulsive forces, since the 
principle of Fermat gives the same laws as that of the least ac- 
tion, by reversing the expression of the velocity. 
