RADIATION  OF  HEAT  ON  THE  PROPAGATION  OF SOUND.     151
that the radiation takes place as if the air heated by compression radiated into an infinite medium at a temperature 0_0. Of course the same reasoning will apply to the apparent radiation of cold. Hence the formula (13) may be applied without change to the vibration of the air within a long tube, and accordingly may be employed in considering the experiments of M. Biot above alluded to.
The preceding view of the effect of radiation within a tube is very different from that taken by M. Poisson in his Traite de Mecanique (vol. ii. art. 665). The latter, however, is contained in a mere passing remark offered by way of conjecture, and probably written without much consideration, and therefore ought hardly to be regarded as supported by Poisson's authority.
Let us now pass to numerical values, in order to make out, independently of any assumption respecting the true explanation of the velocity of sound, whether q must be regarded as very great or very small compared with n. It follows from (13) that the decrease of intensity in going one wave's length in the direction of propagation is a maximum when tan-\|r, and therefore yfr, is a maximum. Now (12) shows that yfr is a maximum when
(16),
K being the quantity defined by (14).    For the above value of q we get from (11) and (12),
^= •/i/d-ijK'-l,    2^= tan-^1 7^4 - tan"^1^-^ ...... (17).
The velocity of propagation, which is equal to n/xT^1 sec ^, docs not much differ from npT^1, since ty, as will immediately appear, is not very large. It may bo observed, that the expression for nfjT^1 given by the first of equations (17), is a geometric mean between the velocities of propagation resulting from the theories of Newton and Laplace.
The value of K, deduced from experiments in which the theory of sound is not assumed, is about 1*36*, whence 2ty = 8^n 47'. If
* Poissou, Traite da Ah'caniqiie, vol. ii. art. G37. The value deduced irom the observed velocity of sound is somewhat larger, and is more likely to bo correct. I have employed the value 1-30 in order to avoid arguing in a circle, because I am reasoning as if the received theory of sound were not established.