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VIII. On the Mathematical Theory of Sound. By the Rev. S. Eaenshaw, M.A., 
Sheffield. Communicated hy Professor W. H. Millee, F.R.S. 
Eeceired Norember 20, 1858, — Bead January 6, 1859*. 
L\ making certain investigations on the properties of the sound-wave, transmitted 
through a small horizontal tube of uniform bore, I found reason for thinking that the 
equation 
l=KI) ■ • 
must always be satisfied ; F being a function of a form to be determined. Differen- 
tiating this equation with regard to t, we find 
dC [ \dx) j dx^' 
( 2 .) 
which by means of the arbitrary function F can be made to coincide, not only with the 
ordinary dynamical equation of sound, but with any dynamical equation in which the 
in terms of 
dx 
ratio of ^ and ^ can be expressed 
Equation (1.) is a partial first integral of (2.), and by means of it we shall be able to 
obtain a final integral of (2.), which will be shown to be the general integral of (2.) for 
wave-motion, propagated in one direction only in such a tube as we have supposed, by 
its satisfying all the conditions of such wave-motion. 
It will be convenient to begin with the simplest case of sound, — that in'which the 
development of heat and cold is neglected. 
I. waye-motiojn when change of tempeeathee is neglected. 
1. The equations for this case of motion are, the dynamical equation 
/^V^_ 
\dxj df- ^ dx^' 
and the equation of continuity, 
^_£p; 
dx q' 
(3.) 
( 4 .) 
fo, Po ‘‘■re the equilibrium density and pressure at any point of the fluid ; p the same 
for a particle in motion ; x the equilibrium distance of the same particle from a fixed 
plane cutting the tube at right angles ; and t is the time when the same particle, being 
in motion, is at the distance y from the same plane ; y, is the constant which connects f 
and by Boyle’s lawp=|!Ao. 
* Subsequently recast and abridged by the author, but without introducing new matter. 
MDCCCLX. T 
