Page:Aerial Flight - Volume 1 - Aerodynamics - Frederick Lanchester - 1906.djvu/435

Then the value of $l^{3}$ for each element will depend upon the number of elements into which the enclosure is divided, so that $l^{3}\propto {\frac {1}{n}},$ and thus

$P_{1}=k\ n\ h_{1}$

$P_{2}=k\ n\ h_{2}$

$P_{3}=k\ n\ h_{3}\ {\mbox{etc.}}$

where $k$ is a constant.

∴	$P_{1}+P_{2}+P_{3}+{\mbox{etc.}}=$ $k\ n\ (h_{1}+h_{2}+h_{3}+{\mbox{etc.}})$	⁠
But	${\frac {P_{1}+P_{2}+P_{3}+{\mbox{etc.}}}{n}}=P_{m}$
and	$h_{1}+h_{2}+h_{3}+{\mbox{etc.}}$ is total heat added $=H$
∴	$P_{m}=k\ H$

this result continues to apply when the number of elements $n$ becomes indefinitely great, hence the proposition is proved.

 

ADDENDUM D.
A Retrospective Note.

It is perhaps of some interest to state that the investigations included in the present appendix were actually made in the early part of 1905; the portion relating to the theory of sound momentum was submitted in the form of a draft paper to Professor Poynting, then President of the Physical Society, with whom the author had some correspondence on the subject.

The author did not receive sufficient encouragement to think it worth while submitting the paper, especially in view of previous experience and of the fact that not only Poynting, but Larmor, and at that time Rayleigh, were thoroughly identified with the general doctrine of sound momentum.

Referring to a warning note raised by the author, and with regard to the suggested paper, Professor Poynting wrote on June 9th, 1905, “Yes, I am quite sure about my views. But it is quite evident that we are not going to see in the same direction. I shall probably send my proof of pressure to the Physical

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