Page:Elementary Principles in Statistical Mechanics (1902).djvu/123

Similar formulae may be used to derive $V_{q}$ or $\phi _{q}$ for the compound system, when one of these quantities is known as function of the potential energy in each of the systems combined.

The operation represented by such an equation as

$e^{\phi _{12}}=\int e^{\phi _{1}}e^{\phi _{2}}\,d\epsilon _{1}$

is identical with one of the fundamental operations of the theory of errors, viz., that of finding the probability of an error from the probabilities of partial errors of which it is made up. It admits a simple geometrical illustration. We may take a horizontal line as an axis of abscissas, and lay off $\epsilon _{1}$ as an abscissa measured to the right of any origin, and erect $e^{\phi _{1}}$ as a corresponding ordinate, thus determining a certain curve. Again, taking a different origin, we may lay off $\epsilon _{2}$ as abscissas measured to the left, and determine a second curve by erecting the ordinates $e^{\phi _{2}}$. We may suppose the distance between the origins to be $\epsilon _{12}$, the second origin being to the right if $\epsilon _{12}$ is positive. We may determine a third curve by erecting at every point in the line (between the least values of $\epsilon _{1}$ and $\epsilon _{2}$) an ordinate which represents the product of the two ordinates belonging to the curves already described. The area between this third curve and the axis of abscissas will represent the value of $e^{\phi _{12}}$. To get the value of this quantity for varying values of $\epsilon _{12}$, we may suppose the first two curves to be rigidly constructed, and to be capable of being moved independently. We may increase or diminish $\epsilon _{12}$ by moving one of these curves to the right or left. The third curve must be constructed anew for each different value of $\epsilon _{12}$.