THE EEY, S. EAENSHAW ON THE MATHEMATICAL THEOEY OE SOUND. 145 
and that of art. 11 the following — 
36. From (16.) it appears that the velocity of transmission of the front and rear of 
either a positive or negative wave is ; but the velocity of transmission of that part 
of the wave of which the density is is, for a positive wave, 
v/^+^U; 
and for a negative wave, 
k-\-\ 
The part of these expressions to which the bore is due is the term U ; and as k is 
known to be greater than unity, this is greater than U ; and consequently change of 
temperature hastens the fonnation of a bore, and also renders the property of art. 16 
inapplicable here. 
37. As in the case of a negative wave the equation (15.) involves a negative term, it 
is manifestly possible for the piston, in generating a negative wave, to move so quickly 
as to leave a vacuum behind it. The least velocity with which this can happen is 
2 Vkfj, 
k-\ ’ 
which for common air is about 5722 feet per second. But it is necessary to notice, that 
in this and similar extreme results, we are hardly justified in supposing k to be constant 
up to such high velocities. 
38. The expression is a maximum (see art. 24) when 
2 \/k[t. 
T+T’ 
which in the case of common air is equal to about 904 feet per second; and the corre- 
sponding density is 2 
f=(Fn)*"f- 
2 
or, for common air, about 
Hence no gas can rush through a pipe faster than at the rate of 
fr +1 
cubic feet per second. 
39. The change of temperature due to the transmission of a wave through an elastic 
medium has been taken account of, by assuming a law different from that of Boyle, to 
connect pressure with density (art. 32). 
If we generalize the law by assuming 
?o\ 
