# Scientific Papers of Josiah Willard Gibbs, Volume 2/Chapter II

II.

ON THE FUNDAMENTAL FORMULA OF STATISTICAL MECHANICS, WITH APPLICATIONS TO ASTRONOMY AND THERMODYNAMICS.

[Proceedings of the American Assodatian for the Advancement of Science, vol. xxxiii. pp. 57, 58, 1884.]

(abstract.)

Suppose that we have a great number of systems which consist of material points and are identical in character, but different in configuration and velocities, and in which the forces are determined by the configuration alone. Let the number of systems in which the coordinates and velocities lie severally between the following limits, viz., between

 ${\displaystyle x_{1}}$ and ${\displaystyle x_{1}+dx_{1},}$ ${\displaystyle y_{1}}$ and ${\displaystyle y_{1}+dy_{1},}$ ${\displaystyle z_{1}}$ and ${\displaystyle z_{1}+dz_{1},}$ ${\displaystyle x_{2}}$ and ${\displaystyle x_{2}+dx_{2},}$ etc., ${\displaystyle {\dot {x}}_{1}}$ and ${\displaystyle {\dot {x}}_{1}+d{\dot {x}}_{1},}$ ${\displaystyle {\dot {y}}_{1}}$ and ${\displaystyle {\dot {y}}_{1}+d{\dot {y}}_{1},}$ ${\displaystyle {\dot {z}}_{1}}$ and ${\displaystyle {\dot {z}}_{1}+d{\dot {z}}_{1},}$ ${\displaystyle {\dot {x}}_{2}}$ and ${\displaystyle {\dot {x}}_{2}+d{\dot {x}}_{2},}$ etc.,

be denoted by

 ${\displaystyle L}$⁠${\displaystyle dx_{1}}$⁠${\displaystyle dy_{1}}$⁠${\displaystyle dz_{1}}$⁠${\displaystyle dx_{2}}$⁠etc.⁠${\displaystyle d{\dot {x}}_{1}}$⁠${\displaystyle d{\dot {y}}_{1}}$⁠${\displaystyle d{\dot {z}}_{1}}$⁠${\displaystyle d{\dot {x}}_{2}}$⁠etc.
The manner in which the quantity ${\displaystyle L}$ varies with the time is given by the equation
 ${\displaystyle {\frac {dL}{dt}}=\textstyle -\sum \displaystyle \left[{\frac {dL}{dx}}{\dot {x}}+{\frac {dL}{d{\dot {x}}}}{\ddot {x}}\right],}$
where ${\displaystyle t,x_{1},y_{1},z_{1},x_{2}}$, etc., ${\displaystyle {\dot {x}}_{1},{\dot {y}}_{1},{\dot {z}}_{1},{\dot {x}}_{2}}$, etc., are the independent variables, and the summation relates to all the coordinates. The object of the paper is to establish this proposition (which is not claimed as new, but which has hardly received the recognition which it deserves) and to show its applications to astronomy and thermodynamics.