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.

266 UNDULATORY THEORY OF OPTICS.
PROP. 9. To explain the reflection of light on the
undulatory theory.
31. We shall again refer to the motion of sound for
an analogical illustration of this point. In fig. 7 let ABCD
be the front of a wave (which for simplicity we suppose
plane, every part moving in parallel directions) advancing
in the direction BB' or CC" and meeting the smooth wall
C"B'. Then it appears from the investigation of sound*
* Let x, y, z, be the original co-ordinates of any particle of air: and at
the time t let them be x+ X, y + Y, z + Z. Then the particle which originally
had for co-ordinate x + 6x will at the time t have
dX
x + x +X + ex nearly;
dz
or the distance between two particles in the direction of x, which was originally
x is, at the time t, ex 1 + d) nearly. Similarly the distances in the directions of y and z which were originally èy and 3z are, at the time t,
S ( dY) and Sz ( dZ\)
Consequently, the air which occupied the rectangular parallelopiped whose sides
were ex, 8y, %z, now occupies the parallelopiped, nearly rectangular, whose
sides are
x 1+dX dX, y1+, ) z Y\, d )
And if the elasticity (represented by the pressure upon a unit of surface) was
originally P, and varied as (density)", (m being nearly -, the elasticity of the
air in this parallelopiped is nearly
dX dY dZ\
P 1 - -m - n I.
dx dy d/
This then is the expression for the elasticity of the air about that point whose
co-ordinates were originally x, y, z: the alteration of elasticity being supposed
small.
Consequently, at the time t, the elasticity about that point whose co-ordinates
were originally x + h, y, z, is
dX d Y dZ\ d f dX dY dZ
P i-m ---m ---m — +P- -m h.
dx dy y dz dx dx dy dz
And therefore if there is a small parallelopiped whose sides were h, k, 1, the
excess of pressure which urges it on in the direction of x is
d ( dX dY dZ
-_p 1P- — m -d- h.kl
dx dy dz
=. Ph kl dz d dx dy + d z )