Page:Scientific Memoirs, Vol. 1 (1837).djvu/18

If between the flame of a candle and the eye we interpose a plate of glass or any other substance more or less transparent, we find the diminution of the intensity of the light always the same, however the distance between the plate and the candle may vary. The effect produced by distance on the freely transmitted caloric is exactly similar; and if at a certain distance from the active source there be a thermoscopic apparatus sensible to this portion of heat, the apparatus will always give the same indication, whether the screen be laid close to the source or to the thermoscope.

But it is clear that it must happen quite otherwise to the conductible caloric; for this portion of the heat, when it has reached the further surface of the screen, leaves it in the form of diverging rays which become weaker in proportion to the distance. In other words, the further surface of the screen being heated becomes a new calorific source whose intensity of radiation must decrease as the distance increases.

We possess, therefore, a very simple contrivance for destroying the influence of conduction, if we keep the action of the free radiation intact. This contrivance consists in removing the screen so far from the thermoscope that the radiation of its own heat may, on account of its extreme feebleness, be totally disregarded.

There are, however, some precautions to be taken; for in proportion as the distance between the screen and the thermoscope is increased, the distance between the source and the screen is diminished. The latter is therefore more heated, and radiates with greater force upon the instrument. It is easy to show by calculation that we always gain; that is, that we always weaken the conductible caloric more and more by removing the screen from the thermoscope, until we have placed it midway between the thermoscope and the source^[1]. Let us, therefore, put the screen in this position (which is the most favourable of all), and we shall see that its heat has then no appretiable influence on the re-

1. Let $a$ be the distance from the source to the thermoscope, $x>$ the distance from the thermoscope to the screen, $i$ the calorific intensity of the source, we shall have ${\frac {i}{(a-x)^{2}}}$ as the expression for the radiation which strikes the anterior surface of the screen. This quantity will become ${\frac {ci}{(a-x)^{2}}}$ at the further surface, $c$ being a constant quantity depending on the conducting power of the matter of the screen. In fine, the radiation of the further surface on the thermoscope will be expressed by ${\frac {ci}{x^{2}(a-x)^{2}}}$; its minimum $(y)$ is to be determined. Now, by differentiating we obtain ${\frac {dy}{dx}}={\frac {2ci(2x-a)}{x^{3}(a-x)^{3}}}$; the equation which gives the quantity will then be $2x-a=0$, whence $x={\frac {a}{2}}$.