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.

ACCELERATION OF PHASES BY TOTAL INTERNAL REFLECTION. 347
sin i. cos i - u sin i \/(- 1) ^/(pY sin i - 1)
sin i. cos i + sin i (- 1)v (g sill i - 1)'
or coso - /(- 1). sin 2,
where tan — 2s i -
cos i
It is improbable that these formule are entirely without
meaning: what can their meaning be?
134. M. Fresnel seems to have considered that as the
direction of the reflected ray and the nature and intensity
of the vibration were already established, there remained but
one element which could be affected, namely, the phase of
vibration. And it seems not improbable that this may be
affected, inasmuch as the incident vibration, though it cannot cause a refracted ray, must necessarily cause an agitation
among the particles of the ether outside the glass. It would
seem to us most likely that the ray would be retarded
(though the phenomena to be hereafter described compel us
to admit that it is accelerated): and in all probability differently according to the direction in which the vibrations
take place. Nothing then seems more likely than that 20
and 2 should express these accelerations*: and as they are
* M. Fresnel's reasoning is of this kind. In several geometrical cases, the
occurrence of an impossible quantity indicates a change of 900 in the position of
the line whose length is multiplied by \/-i. It is probable then that here the
multiplication by \/(-1) denotes that the phase of the vibration which it affects
is to be altered (suppose increased) by 900. Thus the expression
{cos 2 0 + (-1). sin 20}. sin - (vt-x)
is to be interpreted as signifying
cos20. sin (v t- a) + sin20. sin - (v t -+ 90~),
À X
or cos20.sin -- (vt- x)+ sin20.cos - (vt- x),
7 X
or sin (vt- )+20.
And similarly for the other.