Mathematical tracts on the lunar and planetary theories, the figure of the earth, precession and nutation, the calculus of variations, and the undulatory theory of optics.

342 UNDULATORY THEORY OF OPTICa.
These formulae apply equally to refraction from air into
glass, and from glass into air, giving i and i' their proper
values. The intensities of the rays will be represented by
the squares of the coefficients.
PRoP. 26. Light polarized perpendicular to the plane
of incidence falls on a refracting surface: to find the intensity of the reflected and the refracted ray.
129. We cannot here use tle same kind of reasoning
as in (128), because the motion of displacement (being in
the plane of incidence and perpendicular to the path of the
ray) is not in the same direction for any two of the three
rays. To overcome this difficulty, M. Fresnel has adopted
the following hypotheses. First he supposes that the law of
vis viva holds: that is, that the sum of the products of
each mass by the square of its velocity is constant. (This
is certainly true if as in (128) masses are supposed to act
nearlv as elastic bodies. And in all cases of mechanical
action it is equal to the sum of all the integrals of force x
space through which it has acted, wliich is constant in all
the cases of undulation that we can strictly examine, and is
probably constant in this). Next he supposes that the resolved parts of the motion perpendicular to the refracting
surface will preserve after leaving the surface the same relation which they have there, and which, if they follow
the same laws as those of the impact of elastic bodies,
would be thus connected: the relative motions before and
after impact will be equal in magnitude but opposite in
sign. (This is confessed by M. Fresnel to be purely empirical). Adopting these hypotheses, and considering the
masses to be as
sini'. cosi: sin i.cosi',
and representing the displacements in the incident, refracted,
and reflected ray, (estimated positive in that direction perpendicular to their respective rays which is nearest to that
of a body falling perpendicularly front vacuum on the refracting surface,) by a, b, c, we have the following equations:
*i*'. * 2 *. *.. * *
sin i'. cos i. a2 = sin i. cosi'. b2 sin i'. cos i c
a cos i = b cos ' + c. cos i.