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

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