144 AN EXAMINATION OF THE POSSIBLE EFFECT OF THE
cooling referred to a difference of temperature unity. We have, therefore,
M = pto-q0........................(4).
The six general equations (1), (2), (3), (4) serve, along with the equations of condition relating to any particular problem, to make known the six unknown quantities u, v, w, p} s, 6.
To simplify the question as much as possible, I shall take the case of plane waves. Taking the axis of x perpendicular to the planes of the waves, we have v = 0, w = Q, and u, p, s, 6 will be functions of only x and t. The equations (1) and (2) become
dp du ds du A ,„.
—— =—n ---- --—\~ —- = 0 ...............(5):
rlw ' df fJi r/v ^ ' *
\J(jdj U/U U/U \Jbib
and eliminating p and u from these equations and (3), we get
.....................(6).
Eliminating 6 between (4) and (6), we get
(d \d*s 7 f/- -. d } d2s /Krx
_ _l_ q _ — A?-{(1 4-ap)^T 4- Qr -7-2 .........(7).
\ctt j ciu [ at j dx
This equation is satisfied by
8 = A'#»w...........................(8).
where A' is an arbitrary constant, real or imaginary, and m', n are two real or imaginary constants connected by the equation
.....................(9)-
If we suppose mf wholly imaginary, the formulae will refer to an infinite series of waves, the expressions for s} &c., involving x under the circular functions sine and cosine. In this case our formulae would make known the manner in which the motion alters with the time. If we suppose n' wholly imaginary, the motion will be periodic as regards the time. In this case we must not suppose the fluid unlimited, but bounded in one direction by a vibrating plane which keeps up the motion. I shall select the second case for consideration, inasmuch as it is analogous to that of the vibrations propagated along a long tube from a sonorous body at one end of it, and accordingly will bear on the